Hello everyone, my name is Dr. Seema. I'm going to present a few cases of Kono Trunkle Abnormalities. I work as a radiologist in Janani Scanning Centre. My co-author is Dr. Anand Sagar K. Patil. Kono Trunkle Abnormalities are a broad category of congenital heart diseases that include a variety of conditions like tetralogy of fallow and its variants like absinth pulmonary wall and tetralogy of fallow with pulmonary atresia, double alkyl trite ventricle and its variants, transposition of great arteries and persistent rancous artery uses. These are relatively common anomalies in postnatal series comprising about 10 to 12% of all the congenital heart defects diagnosed after birth. However, the prenatal diagnosis of these anomalies is more challenging since most of these anomalies have a normal fourth chamber view during routine fetal screening. The prevalence in reported series varies from 2.5 to 21% with a diagnostic accuracy of 75 to 90% for achieving a complete diagnosis. The aims and objectives of our study was to identify the fetuses with Kono Trunkle Abnormalities through trans-abdominal ultrasound using standard planes as per ISOG guidelines to study imaging features of Kono Trunkle Abnormalities. So the materials and the methods used were the fetuses were examined trans-abdominally in all antenatal cases between 70 to 26 weeks of gestation, standard axial planes were taken and analyzed, complementary sagittal and parasagital views were also acquired in suspected cases where we had to look at the iotic arch, ductile arch and the bicable view. Findings were confirmed by prenatal autopsy and or review of images by the experts. 763 fetuses between the gestational age of 17 weeks to 26 weeks were examined in the period of 24 months from June 2019 to June 2021. 20 fetuses were diagnosed with structural defects consistent with Kono Trunkle Abnormalities out of which we are presenting 10 cases. Diagnosis was confirmed by post-mortem examination or expert review. With careful survey of atrial ventricular and ventricular arterial concordance, we could diagnose 5 cases of 20 weeks at 20 weeks, 1 at 19 weeks, 2 cases at 21 to 22 weeks, 1 at 17 weeks and 1 at 26 weeks. History of consign unity was present in two cases. All of them were terminated. We could convince one patient for autopsy and confirm the findings. Now coming to the case one. The clinical history, 30-year-old Prani came for an anomaly scan at 20 weeks. Second degree consign unity was present and no significant family history was present. The ultrasound findings where there was no, in the outflow tract views, we could see that there was no crossover. Parallel course of two arteries was seen. Diota was arising from the anterior right ventricle, but as a pulmonary artery is connected posteriorly with the left ventricle. Ventricular arterial discordance was found whereas there was atrial ventricular concordance with right ventricle connected to the aorta and the left ventricle connected to the pulmonary artery. In three vessel trachea view, we could see only two vessels with a typical eye sign and all these findings were consistent with transposition of great arteries. Coming to the images, in this image we confirmed that there was cytosolitis. The fetus was in cephalic presentation, so the vertebral body, this funding bubble and the apex all should be in clockwise direction as it was so. So it was a cytosolitis. We confirmed the cytos. Next thing what we did was we had to confirm the cytos of the heart. That is the right side was the right side, that is the right atrium was draining into the right ventricle and the left atrium was draining into the left ventricle. So we confirmed that there was atrium ventricular concordance but in the outflow tracts we could visualize the paralleling. And the careful examination of the outflow tracts showed us that there was a non-branching vessel, non-branching vessel which was arising from the anterior ventricle that was the right ventricle and that made us conclude that it was the aorta which was arising from the right ventricle. And in the three vessel trachea view we could see only eye sign. Only one vessel was seen. That was the superior vanakeva and the single large vessel was seen in three vessel trachea view. So we confirmed the diagnosis that it was a complete transposition of great arteries. Coming to our next case it was a 25 year old primie who came to us for a routine scan at 26 weeks. In non-consanguous marriage there was no significant family history. The ultrasound findings where the connections of the systemic and the pulmonary veins was normal that was the pulmonary veins were draining into the left atrium and the inferior superior vanakeva were draining into the right atrium. But when we went on to see the atrium draining into the ventricle what we found was an abnormal connection of the atria and the ventricle that is atrioventricular discordance. And the position of the ventricles was reversed. The morphological right ventricle was seen connecting the left atrium and the morphological left ventricle was seen connecting the right atrium. So the ventricular anatomy showed us there was a atrioventricular discordance. Minimal fluid was seen in the pericardium and the great vessels when we had a look at the great vessels there was no crossover and both the arteries were following a parallel cross course instead of the criss-crossing. So the aorta was arising from the anterior right ventricle and the pulmonary artery was arising from the left ventricle. This findings that's consistent with corrected transposition of great arteries. Let's have a look at the images. So first of all it was a breach presentation. So we expect the vertebral body, stomach and the apex to be in the anticlockwise direction. This is cytosolitis. So the cytos was confirmed, visceral cytos was confirmed. But when we had a look at the cytos of the heart, the left atrium was seen connecting to the right ventricle and the right atrium was seen connecting to the left ventricle. So how did we come to this conclusion was the left atrium was being separated from the lower ventricle by tricuspid wall and there was a moderator band in the ventricle. So we concluded that it was a right ventricle. Now when we concluded that this was a right ventricle, then we went on to look at the outflow tracts. The morphologically looking left ventricle which was draining the right atrium was being drained by the main pulmonary artery which was a dividing vessel. But we know that left ventricle gives rise to a non-branching vessel that is a iota. So here we found that there was double discordance. So double discordance is always consistent with the finding of corrected TGA. And in the sagimages we could see that the iotic arch was arising from the right ventricle and the ductus arch was just beneath it and this confirmed our diagnosis that there is a parallel orientation of the great arteries because of the double discordance there is also corrected TGA. This image shows us that the pulmonary artery, the main pulmonary arteries arising from the left ventricle and giving its branches. The pulmonary artery was normal in its caliber. So this is a 3D image which confirms all our findings. This is a right atrium draining into the smooth wall left ventricle and that is being drained by the pulmonary artery. This is a left atrium draining into the right ventricle and further draining into the iota. So this confirmed our diagnosis. So this is a case 3, 29-year-old Primi who came to us at 22 weeks for an anomaly scan, non-consanguineous marriage, no family history. Here in this case we could find that the position of the ventricles was normal, atrial connections were normal, ventricular and anatomy showed us a 3.6-millimeter size inlet BSD. Great arteries, both the outflow tracts were seen arising from the morphological right ventricle with anterior, large iota and posterior pulmonary artery. In three-veginal trachea view we could see a single large vessel which was iota. Atrial ventricular septum we could see a pulmonic BSD. Ductal arch and iotic arch were seen paralleling. So this made us think about tau-sigbing form of double outlet right ventricle with sub-pulmonic and inlet BSD. So coming to the images. So this was a breach presentation. Again the vertebral body, the stomach and the apex all are on the same side or in the anti-clock direction and this confirms the phytosolitis. The right atrium it was raining into the right ventricle, left atrium into the left ventricle. We could find a inlet BSD there. Then after passing on to the outflow tract we could see that there was a BSD just beneath the great vessels which was beneath this pulmonary artery. And this was called a sub-pulmonic BSD. We could see that the right ventricle was giving rise to the iota as well as the main pulmonary artery. There was paralleling of the arches as seen in this figure, iotic arch and the ductal arch. Iotic arch arising from the anterior ventricle that was the right ventricle. Here in this image you can clearly see the origins of the outflow tract arising from the right ventricle. That concludes us to the diagnosis of tau-sigbing type of double outlet right ventricle. Double outlet right ventricle are of four types depending upon from where exactly are the blood vessels arising. Here in the most important things which we should look at are the relation of the great vessels, the position of the BSD, presence or absence of pulmonary pulmonary stenosis, absence of all mitraiotic and mitra-pulmonary continuity. The diagnosis of DORV is important because they are associated with trisomy 18 and 13. Then the types, DORV with subbiotic VSD, DORV with sub pulmonic VSD, doubly commuted VSD and remote VSD. These are the four kinds of DORVs. Coming to the case for 30 years old primary came for a routine scan at 20 weeks and non-consangular marriage, no family history. Poor chamber showed mild symmetry with right ventricle predominance. The aorta is dilated and overriding. There is a septoietic discontinuity and we could also find a malaligned VSD of 1.9 millimeters. The pulmonary artery in this case was normal. Our diagnosis was isolated finding which is malaligned VSD with overriding of aorta. So this is a four chamber, this is a five chamber view which shows us that the aorta is overriding. It is doubly commuted to the left and the right ventricle. Rest all findings were normal. The pulmonary artery was normal. If the pulmonary artery was smaller, we could have diagnosed it to be a tetralogy of fallow but that wasn't the case. Case number five, 27 year old primary came for a normally scan, non-consangular marriage, no family history. Here what we could see is the cardiac apex was centrally placed that was mesocardia and the great vessels were abnormal. Ventricle arterial discordance was found. Aorta was arising from the right ventricle. Pulmonary artery in this case was erratic. On three vessel view, we could see a single large vessel that was a aorta to the left of the superior vena cava. AVSD was noted. The apex in the center, the atrial ventricular discordance was maintained. The vessels arising from the interior chamber did not show bifurcation that confirmed that it was aorta but it was giving Nick vessels. So this suggested the diagnosis of PGA and in this case the pulmonary artery was smaller so we gave diagnosis of complete transposition of great arteries with atrial ventricular septal defect and pulmonary arteries. These are the findings. There is mesocardia. There is atrial ventricular discordance. There is AVSD and this is the right ventricle giving rise to a non-branching vessel which is aorta. In three vessel, we have a single vessel to the left of the aorta. This is a SAGE view showing us that the aorta is arising from the right ventricle. This is a AVSD. Case 6, 25-year-old tiny anomaly scan at 20 weeks, non-consanguous marriage, no significant family history. On examination of the heart, we could find that there is left-axis deviation. There was 2.2 millimeter perimembranus VSD noted. Aorta and pulmonary artery are seen arising from the right ventricle. Pulmonary artery appeared narrow compared to the aorta. Three vessel trachea view shows only two vessels. Thymus was also absent. This is a complex cardiac case of DO-RV with atritic pulmonary artery and right-sided aortic arch. This is a perimembranus VSD here. This slide shows us there are only two vessels at 3V-TV and the aorta is crossing on the right side of the trachea. This is the right-sided aortic arch. This is to show that the pulmonary artery and the aorta both are arising from the single ventricle, that is the right ventricle. This is case 7. 31-year-old G2P1 came for anomaly scan, second-degree consanguinity, no family history. Cytus was normal. There was septo-iotic discontinuity measuring 4.5 millimeters. Aorta was dilated and overriding. Three vessel trachea view showed that the aorta appeared dilated and short forward flow. This was a case of tetralogy of fallow. This confirms the atrium ventricular concordance from the four chamber view. The five chamber view shows that the aorta is a septo-iotic discontinuity. There is a septo-iotic and discontinuity very nicely shown here. As compared to the aorta, the pulmonary artery was looking very narrow. This made us conclude that it was a case of tetralogy of fallow. There are a few findings on atopsi which was confirmed. There was overriding of aorta. There was ventricular septal defect. There is small, tiny vessel which is in your pulmonary stenosis. Case 8, 18-year-old primary who came to us at 17 to 18 weeks for early anomaly scan in non-consanguinous marriage and no significant family history. Here we could find a single vessel raining both the ventricles committed to more to the left ventricles such as your common arterial trunk, bar, pulmonary atria with VSD. These are the findings showing a single large vessel committed to both the ventricles. This is a color flow which confirmed which tells us that there is only a single vessel raining both the ventricles. Case 9, it was a 21-year-old primary who came for an anomaly scan at 19 weeks, non-consanguinous marriage and no family history. On ultrasound finding, we could find that the dilated right ventricle with perimembranous ventricular septal defect measuring 2.5 millimeters on the right ventricle. When we had a look at the outflow tracks, we could find that the pulmonary artery there was bulbous dilatation of the main pulmonary artery as well as the right and left branches just proximal and distal to the dysplastic pulmonary walls. After cut-off of the pulmonary artery we noted the level of ductus arteriosus due to ductus arteriosus absence. In 3-vessel trachea view, we could find the bulbous dilatation of both the pulmonary arteries and both the branches of the pulmonary artery and the main pulmonary artery. So this made us conclude that there was absent pulmonary, absent pulmonary valve syndrome. So this is a cytosolitis and this is the bulbous dilatation of the pulmonary artery giving rise to the dilated bulbous dilatation of the pulmonary artery branches. So this is the color showing there is a turbulent flow at the level of the pulmonary walls and regurgitation as well. So there was a BSD and this is a right-sided iotic arch and this is the main pulmonary artery this is the dysplastic pulmonary this is the RPA and the LPA this was a case of absent pulmonary valve. The absent pulmonary valve that is leads to post-absent pulmonary wall or abnormal pulmonary wall leads to post-genotic dilatation of main pulmonary artery and its branches. The types are non-top and top type. This was a top type of in our case it was a top type of APVS. There are four important features that is the ballooning of the main pulmonary artery and its branches. Second one is the regurgitation at the pulmonary wall and we find high velocities across the wall. Case 10 24-year-old Primi who came to us for anomalies can at 21 weeks non-consummonous marriage, no family history both the atrium are opening into a single morphological left ventricle at right side was hypoplastic and a left-sided morphological left ventricle was communicating through a VSD and great vessels there was no crossover again and there was a parallel orientation and the iota was arising from the anterior right ventricle whereas the pulmonary artery was connected posterior to the left ventricle. So the ventricular atrial discordance at the rudimentary right ventricle connected to the iota and the left connected to the pulmonary artery. Iota is anterior and right wall position related to the trachea was the only two vessels in the 3VTV and the ventricular connections was reversed coming to the final diagnosis was double inlet left ventricle with anterior and rightward iota. So here we can see that the left atrium and the right ventricle and the left atrium both are draining into the morphological left ventricle right ventricle was hypoplastic, the cytos was maintained. So this is the tricuspid wall this is the mitral wall which has seen opening into the left ventricle on 3VTV two vessels were seen which was the iota. So this is the inlet VSD there is a malposition LBOT was arising from the morphological hypoplastic left ventricle and the RBOT was arising from the left ventricle so the ductus arch arising from the left ventricle and the iotic arch arising from the right ventricle and the pulmonary artery was arising from the left ventricle. So this is the table showing the findings of each case and the outcome and confirmation was done by what the first case was transposition of great arteries confirmed by expert opinion went for termination corrected TGA confirmed by expert opinion they went for termination complex cardiac disease with toxic bing form of DORV with inlet VSD and supalmonic VSD confirmed by expert opinion fourth case was just overriding with malaline VSD termination complete transposition of great arteries with AVSD termination DORV with atritic pulmonary artery and right sided iotic arch terminated tetralogy of fallow went for they went for termination then autopsy and I have shown the images common arterial trunk and pulmonary artery with VSD they just terminated it absent pulmonary valve termination and DILV they went for termination con truncal abnormalities and structural abnormalities involving the outflow tracts the diagnosis becomes difficult here because most of them have normal four chamber view so detailed and extended cardiac examination including the ventricular arterial and the atroventricular connections is a paramount importance so the fatal diagnosis of structural heart disease is currently increasing due to the policy of prenatal screening of cardiac malformations during the routine scans this is very important for us to diagnose these cases the conclusion is fetal echo is a valuable tool in early detection of serious defects such as these con truncal anomalies because they require early fetal early fetal or postnatal interventions especially in cases with high risk positive consignority is a major serious problem in developing countries which is usually associated with serious major problems in the offsprings thus early screening to detect these defects that may need early intervention the findings of our study reflect that the careful survey of outflow tracts would help in diagnosis of con truncal abnormalities not only the fourth chamber view the most common errors that are associated with the determination of rate artery relationship as most of the abnormalities had a warlock low socioeconomic status and lack of awareness about the facilities at the tertiary sent as lead to termination in all our cases in our study we had cases of fallow DORV complete TG and corrected TG in the decreasing order so these are my references thank you so much


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Hello and welcome to the session. Let's work out the following question. It says prove the following identity where the angles involved are acute angles for which the expression is defined. So let's now move on to the solution and let's start with LHS which is tan theta upon 1 minus cot theta plus cot theta upon 1 minus tan theta. Now we'll simplify LHS this is equal to tan theta can be written as sin theta upon cos theta upon 1 minus cot theta which can be written as cos theta upon sin theta plus cot theta is cos theta upon sin theta upon 1 minus tan theta and tan theta is sin theta upon cos theta. Now we have sin theta upon cos theta into taking LCM and then simplifying we have sin theta upon sin theta minus cos theta plus here also we'll take LCM here and we'll multiply this with the reciprocal of this. So we have cos theta upon sin theta into cos theta upon cos theta minus sin theta. So this is equal to sin square theta upon cos theta into sin theta minus cos theta plus cos square theta upon sin theta into cos theta minus sin theta is again equal to sin square theta upon cos theta into sin theta minus cos theta minus here we take minus common so we have cos square theta upon sin theta into sin theta minus cos theta now taking LCM so LCM would be sin theta cos theta into sin theta minus cos theta and in the numerator we have sin cube theta minus cos cube theta now again this is equal to here we'll apply the formula of a cube minus b cube which is a minus b into a square plus a b plus b square so the numerator will become sin theta minus cos theta into a square here a is sin theta and b is cos theta plus sin square theta plus a b b is cos theta so it is sin theta into cos theta plus cos square theta upon sin theta into cos theta into sin theta minus cos theta sin theta minus cos theta gets cancelled with sin theta minus cos theta and let's write the formula we have used now this is equal to sin square theta plus cos square theta plus sin theta cos theta upon sin theta cos theta now we know that sin square theta plus cos square theta is equal to 1 so this is 1 plus sin theta cos theta upon sin theta into cos theta is again equal to 1 upon sin theta into 1 upon cos theta plus sin theta cos theta upon sin theta cos theta sin theta gets cancelled with sin theta cos theta gets cancelled with cos theta and we have left with 1 upon sin theta is cos square theta 1 upon cos theta is sin theta plus 1 so this is equal to 1 plus sin theta into cos theta and which is the RHS hence proved so this completes the question on the session bye for now take care have a good day


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Let me recall we are probing this result let W be a finite dimensional subspace of an inner product space V let E be the orthogonal projection orthogonal projection of V on W okay then E is an idempotent linear transformation E is an idempotent linear transformation the idempotent linear transformation of V on to W on to W so we must show that this is an on to a map such that the following hold up first condition is that W perpendicular is the null space of V and so remember in order to write this we must know that E is linear see what we know is that the null space or range space of a linear transformation is a subspace so in order to write this we must know that E is linear okay this is one condition that E must satisfy and more importantly V is the orthogonal direct sum of these two subspaces okay for the proof I remember that we have done E is idempotent and E is linear okay so let me just write this already we have shown that E square equal to E that is E is idempotent and E is linear we must show that V is on to then W perpendicular is null space of V and this okay let us first dispose this off this is straight forward V is on to W that is given any X and W can I find okay given any U and W can I find an X in V such that EX equals U it is a same thing if so let me write like this if X belongs to W then what we know is that EX equal to X so the pre-image is X itself so that E is on to let me just emphasize W see E is a linear operator on V so from V to V it is not an on to map but if you look at the subspace W think of E as a mapping from V to W then it is on to okay that is what this means so it is the orthogonal projection of V on to W we must show that W perpendicular is null space of E okay so let us take we need to show this and then this let us take okay let us say Y or U or Z okay Z belongs null space of V implies E Z is 0 by definition in fact if and only if is not it by definition but E Z equal to 0 if and only if remember that to any fixed vector Z in V E assigns the best approximation the unique best approximation to Z from W so and we also know that condition what is that condition if EX equal to U then X minus U is perpendicular to W this happens if and only if X minus U is perpendicular to W that is I am writing it like this Z minus 0 belongs to W perpendicular that is the same as saying Z belongs to W perpendicular and so we have proved in one single step that the null space of E is W perpendicular okay okay this is just the definition how would how is E defined to any vector Z it assigns the unique best approximation and that best approximation U satisfies X minus U perpendicular to W that is what I have used okay finally this equation we need to show that but that is straight forward we show that V is W direct sum W perpendicular let us take X in V then X can be written as I want it to be in this plus this so I can write EX plus X minus EX let us call this as X 1 and this as X 2 X 1 plus X 2 then X 1 belongs to range of E but range of V is by definition W and X 2 is X minus EX so I want you to look at EX 2 EX 2 is E of X minus EX that is 0 EX minus E square X but E square is E so EX 2 is 0 just now we have shown that null space of E is W perpendicular so X 2 belongs to W perpendicular so for one thing we have shown that V can be written as a sum of W 1 sorry W and W perpendicular we must show that it is a direct sum that is we must show that the intersection is the subspace consisting of the 0 vector alone but that is straight forward. So what we have shown is that V is contained in W direct sum W perpendicular so I can write equality because this is a subspace direct sum is also a subspace so this is equal I am sorry W equals just W plus okay finally if let us say Z belongs to W intersection W perpendicular then by definition the inner product of Z with itself must be 0 because Z is orthogonal to every vector in W and Z belongs to W perpendicular also but this means Z is 0 so the intersection is singleton and so it follows that V is the direct sum of these two subspaces. So remember that there is no restriction on V the restriction is only on W it must be a finite dimensional subspace V could be an infinite dimensional inner product space but if W is not a finite dimensional subspace then this is not true okay if time permits I will provide an example later okay so this is a finite dimensional result okay corresponding to E we have this result the version for F which kind of complements E that version is similar so let me give you that version for the record. So the following is a consequence of the previous result remember the mapping F, F is I minus Z okay and we have seen that F is a projection of V onto of V on W perpendicular but what follows from this theorem is that it is an idempotent linear transformation of V onto W perpendicular this time and what is the null space of F what do you expect the null space of F to be what is F? F is I minus E null space of F is W perpendicular I am sorry just W okay null space of F is range of I minus E okay so let us null space of F is range of E yes that is W so let us quickly dispose this one corresponding to E we defined an F so let W be a finite dimensional subspace of an inner product space V and E denote the orthogonal projection of V on W as before then I minus E is an idempotent linear transformation I minus E is an idempotent linear transformation of V onto W perpendicular such that the null space of I minus E is W so previously we had W perpendicular to be the null space of E that is I am looking at the null space of the orthogonal projection at any stage the null space of I minus E is W the proof is almost there so I will not write down the fact that E is idempotent and linear implies that I minus E is linear in the first place and I minus E we have seen this last time I minus E square is I minus E so it is an idempotent linear transformation I minus E that is the mapping that takes X to X minus E X we have seen last time that it is it is the orthogonal projection of V on W perpendicular that was proved we called it F the mapping F from V to V defined by F of X equals X minus E X is the orthogonal projection of V on W perpendicular but from what we have seen just now it must be onto also similar to E I minus E behaves very similar to E but it is complementary to E so it is an orthogonal projection of V onto W perpendicular so this is done and null space of I minus E equals W is straight forward very similar to what we have done before okay so I am going to leave this last part also okay before so this is kind of the summary of the notion of best approximation how it helps in decomposing an inner product space in terms of a finite dimensional subspace and its orthogonal complement okay so what is important is the notion of best approximation okay but remember that for finite dimensional subspaces the best approximation exists and it is unique that is I mean these two are important for us to do all these things using again this notion we will go to the concept of adjoint of a linear transformation then the notion of unitary operators normal operators finally spectral finite dimensional spectral theorem okay but before that let us look at two examples two numerical examples where I will show how okay where I will try to give you another clue as to what you must expect of the linear transformation E when it is the orthogonal projection okay I asked to this question what is the difference between an ordinary decomposition direct sum decomposition and an orthogonal direct sum decomposition okay you may be able to answer by looking at this example so this is kind of reinforcing what we have done till now. So let us look at this example really two examples let us take R2 for R2 I have the following two decompositions let me call W1 as a subspace that is span of 1, 1 W1 is span of 1, 1 W2 is span of 1, minus 1 and I will write these are row vectors these are row vectors then what is the relationship between these two can I say W2 is W1 perpendicular okay observe that W2 is W1 perpendicular let us look at another pair I will call it Z1 Z1 is span of I will take the same vector as before so Z1 is really W1 Z2 let us say is span of the vector 1, 2 span of vector 1, 2 in the first instance obviously W1 direct sum W perpendicular is R2 in the second case can I say Z1 direct sum Z2 is R2 first answer must be instant answer is yes why these are independent vectors together they form a basis and this vector does not belong to the span of this this does not belong to the span of this so the intersection is single term 0 okay so for the record R2 can be written as W1 direct sum W2 perpendicular and equal to okay let me write like this and R2 equals Z1 direct sum Z2 the intersection is single term 0 and these two vectors together form a basis so any vector can be written as a linear combination of these two so for sum Z1 plus Z2 is R2 is clear Z1 intersection Z2 single term 0 is also clear because these are independent vectors just by inspection W1 direct sum W2 or I will write W1 perpendicular yes okay let us find the maps E in both these cases okay what is the property of the map E that we are going to use the subspaces are given we want to find E what I will do is not find E I will just take E to be a matrix itself remember there is this 1 to 1 correspondence between the matrix of a linear transformation and the transformation itself I will write down the matrix of E relative to the standard basis and see how it looks like by the way can you geometrically differentiate between these two direct sum decompositions? Geometry for the first one it is 1, 1 you can think of the line Y equals X okay that is the line which makes 45 degrees with the positive X axis the line 1 minus 1 okay makes 135 degrees with the positive real axis and observe these two lines are perpendicular these two lines are perpendicular in the second case this is the same subspace but look at Z2 Z2 is not perpendicular the line passing through 1, 2 is not perpendicular to the line passing through 1, 1 okay but then you know that since they are independent the span of these two vectors must be the entire space but remember that the angle between these two lines one passing through 1, 1 the other one passing through 1, 2 are not perpendicular this is crucial in the first case in the first case I want to determine E such that okay let E be the matrix I am using the transformation notation itself to denote the matrix let E be the matrix corresponding to okay the projection of W1 onto W2 and so E is let us say alpha, beta, gamma, delta E must be a 2 by 2 matrix okay it is a linear transformation on R2 E is this to determine E completely I must use a property that E acts like identity on the range of E W1 and it must act like the 0 operator on the perpendicular do you agree range of E is W range of E is W null space of E is W perpendicular okay and over range of E that is if X belongs to range of E then E X equal to X if X belongs to range of E then E X equal to X so E acts like identity on its range E acts like identity on W it acts as 0 operator on W perpendicular 2 conditions but you will get 4 equations in 4 unknowns alpha, beta, gamma, delta so we will determine E completely so let us do that in both the cases in the second case also this time we cannot talk about the orthogonal projection we will just talk about the usual projection it is called an oblique projection oblique projection so I will determine E in the second case such that E acts like identity on W1 that is Z1 and E acts like 0 on Z2 then I will compare these 2 case look at the structure of E how does E look like. So first E X equals X if and only if X belongs to W that is the range of E and so I am using this condition W1 right and E X equal to 0 if and only if X belongs to W perpendicular W1 perpendicular that is W2 in this example range of E is one dimensional W1 is one dimensional and span by the vector 1, 1 say I have not taken an orthonormal basis that does not matter span by W1 is span by 1, 1 so E of that vector must be itself so for one thing alpha, beta, gamma, delta operating on 1, 1 must be 1, 1 1, 1 belongs to W1 in fact that is the only vector in this only independent vector in W1. Yes it is V, V on to yes projection of V on to W1 yes I have this in the back of the mind that this will correspond to W1, W2 yes it is a projection of V on to W1 okay so I have this so this equation gives rise to 2 equations okay this matrix equation tells me alpha plus beta equals 1, gamma plus delta equals 1 okay 2 equations in 4 unknowns 2 more equations will come from the second set of equations E X equal to 0 so all the 4 unknowns can be determined so look at this E X equal to 0 if and only if X belongs to W2 I will take that vector 1, minus 1 equate that to 0 alpha, beta, gamma, delta into 1, minus 1 that must be this time the 0 vector 0, 0 okay that gives rise to these 2 equations alpha plus beta is 0 gamma sorry alpha minus beta is 0 gamma minus delta is 0 so alpha equal to beta equals half gamma equals delta equals half okay so let me write down E in this case are my calculations correct okay let us go to the second case now in the case when R2 is Z1 direct sum Z2 remember that there is I do not know if I have told you this before there is this terminology associated with perpendicular subspaces this E, this E is called the projection of V on to W1 along W2, E is the projection of V on to W1 along W2 that is what you have calculated along W2, E is the projection of V on to W1 along W2, along you can look at the geometry okay what is the geometry let us go back to the best approximation problem say I have a subspace pass through the origin this is X okay this is my subspace W and I want the projection of X on to W along W perpendicular so I must go along perpendicular that is all see this is perpendicular so I reach W along the perpendicular to W from X okay that is the reason it is called along W2 in this case it is the orthogonal projection but if you look at the case Z1 Z2 the second decomposition it is not orthogonal projection but we can still say that E is the linear transformation or the matrix of the linear transformation of R2 on to Z1 along Z2 we can still say along Z2 okay this time it is not an orthogonal projection, this time it is not an orthogonal projection okay so what I have tried to do is I have done the whole thing for orthogonal projections and I am trying to imitate this for the ordinary oblique projection I have taken the property of the orthogonal projection and trying to imitate what it satisfies in the non orthogonal case what do I expect of a projection in the non orthogonal case what do I essentially what do I expect the property of a projection that of course it is E square equal to E and it is a linear transformation okay apart from that what are the important properties range of E is W1 null space of E is W2 this is essentially what I am trying to imitate in the non or the oblique case and I am trying to see what is the difference between the matrices that I get what is the difference between the transformations in these two cases okay. So let us go to this case when I write R2 in this manner again I will do a similar thing so I will use these calculations I want E to act like identity on W1 which is Z1 and EX equal to 0 this time Z2 okay observe Z2 is not perpendicular to Z1 okay identity this equation will remain as it is okay Z1 is the same as W1 but the other one will be 1 2 okay this is the only change the other one will be 1 2. So I get these equations alpha plus 2 beta equals 0 gamma plus 2 delta equals 0 in the case when R2 equals this E is given by the matrix just avoid the confusion I do not want to write 2 E's so E is given by this matrix so what is that this time can someone make the quick calculation alpha is minus 2 beta beta is minus 1 alpha is 2 is that correct beta is minus 1 alpha is 2 the next one I will write without looking at the equations what is the next one it is not the same is it the same or minus of that it is the same okay the other see this will be a rank 1 matrix so the second row will be a multiple of the first row in this case the same as the first row this will be a rank 1 matrix you know why this will be a rank 1 matrix this is also a rank 1 matrix I mean this is not invertible this will also not be invertible okay in this case why is it not invertible no no please come on that is not my question you mean I will give you a 2 by 2 matrix ask you to tell me the determinant I want a qualitative answer E is not if you restrict the function from E to E then it is a function is bijective so then E is a linear transformation from R to R what prevents E from being a bijective linear transformation this is the question pardon why is it not an injective why should it not be that is the question can be expect E to be invertible that is the question can we expect E to be invertible yes that is the only situation that is the only situation only these are the only 2 extremes you expect so the only injective see E square equal to E if E is invertible what happens E is identity the other extreme E square equal to 0 the other extreme is that E is 0 now in one case V is equal to W so W perpendicular is trivial in the other case V equal to singleton 0 so W perpendicular is the entire space V these are the 2 extremes okay invertibility cannot happen for a projection if it happens it is identity identity there is no problem the problem is not there in the first place okay so E can never be invertible okay range of V equal to W so if E is invertible then W is the whole of V so W perpendicular is trivial okay so this is my E in the second case what is the difference between the structures of E is there any difference you can of course verify the D square equal to E in this example as well as in this example okay but what is there is there a qualitative difference between the structures of E in these 2 examples if yes what is it is there something that you can say for the first one which you cannot say for the second one structure looking at the form E transpose is E that is essential E transpose equal to E is in fact another characterization for an orthogonal projection E transpose not equal to E corresponds to the oblique projection so E square equal to E E transpose equal to E and range of E determine E completely E square equal to E E transpose equal to E range of E that is the subspace range of E must be known to me these 3 determine E uniquely in the orthogonal direct sum case in the oblique direct sum case I must know range of E I must know null space of E as well as the fact that E square equal to E in the oblique projection case E square is not equal to E sorry E transpose is not equal to E okay so remember there is another important thing which I hope you have observed which is for the subspace W1 there is a unique subspace which together with W1 forms a direct sum decomposition this unique subspace is unique in the sense that it is perpendicular to W1 okay but for the second setup Z2 is just one complementary subspace of Z1 in place of Z2 I could have taken span 1, 3 I could have taken span 1, 4 all these are complementary subspaces to Z1 okay so in the case of orthogonal projection there is a unique decomposition of the vector space V into W plus W perpendicular in the case of ordinary projection there are infinitely many direct sum decomposition that is given a subspace W there are infinitely many complementary subspaces that those can be obtained by an oblique projection those can be obtained by oblique projection the unique subspace is obtained by the orthogonal projection okay so this is really the essential difference between an orthogonal projection and an oblique projection okay so let us then move on to the next topic. The topic is linear functionals and adjoins, adjoins of linear transformations, linear transformations and adjoints of linear transformations, linear functionals is what I want to discuss first linear functionals and adjoints adjoints of linear transformations the notion of adjoint will generalize a notion of symmetry the operator the operation of taking given A taking the operation A transpose given A doing the operation A star that will be generalized in a inner product space we will call that as the adjoint operation but before that linear functionals okay so what is a linear function? Definition a linear functional definition underline field okay that is a linear function okay. What we will do is first give a representation for linear functions okay this result is easy and this also should immediately remind you of a theorem in functional analysis which you will do a little later. Linear functional and adjoint so first I have the following let V be a finite dimensional inner product space V be a finite dimensional inner product space and F be a linear functional on V so the underlying field is real or complex so F is a linear transformation from V to R or C then the representation theorem says there exists a unique vector Y in V there exists a unique vector Y in V such that the operation of this linear functional is like taking inner products with the vector Y such that the action of F on any vector X is given by the inner product X, Y for all X and V for a fixed Y if it is a linear functional then it arises precisely in this manner okay the proof is just by producing the vector Y okay and we will show that Y is unique okay but this is called a representation theorem that is any linear functional is represented by this inner product the proof is we start with an orthonormal basis and give the formula for Y let us say this is U 1, U 2, etc U n let V be an orthonormal basis of V, V is assumed to be finite dimensional so it has an orthonormal finite orthonormal basis I know F I want to find Y which I must show is unique and satisfies this equation. Define Y by Y equals summation J equals 1 to N F of U J bar U J define Y to be this vector F is known so I can determine Y look at the images of the basis vectors under F take the conjugates and then form this particular linear combination okay we will show that this Y satisfies this equation and show that this Y is unique consider for 1 less than or equal to K less than or equal to N the inner product of U K with Y this is inner product U K Y is this summation J equals 1 to N F U J bar U J the inner product is conjugate linear with the second one so when this comes out it goes with the conjugate double conjugate summation F U J into the inner product U K with U J with J being the running index K has been fixed so when J takes a value K this is 1 all other terms are 0 this is orthonormal basis so the only term that remains is when J is equal to K you substitute here it is F U K so we have shown F U K is U K with Y this is orthonormal basis so for any X this will be true because any X there is a linear combination right is that clear that is if X belongs to V then X can be written as summation J equals 1 to N in fact the coefficients are X U J with U J and so if you look at F X then F X is summation J equals 1 to N F is linear so it is a X U J F U J but F U J is I want to show F X equals X comma Y okay X U J F U J this is should have been better to it does not matter can I just say this is inner product X with Y again very you could have started with X with Y start with inner product X with Y then it is X into that formula F U J goes out and then I must take the inner product of U J with X so I get this so you have this representation for any linear functional F so there is a unique Y uniqueness we have to show suppose there is Z in V such that F X equals X comma Z okay then you can compare these two it follows that inner product X with Z is inner product X with Y this means that okay this means that inner product X with Y minus Z equals 0 for all X and V this means since this is true for all X I can replace X by Y minus Z which means Y equals Z so uniqueness follows immediately okay so every linear functional on a finite dimensional inner product space arises from the inner product with a particular vector observe that this Y belongs to null space F perpendicular which is not coming from the proof but this can be shown Y belongs null space of F perpendicular that is can you see why this is true proof let X belong to null space of F I must show that inner product of X Y is 0 then F X is 0 that is 0 equals F X but F X I know is inner product of X with Y so this means Y is perpendicular to null space of F if X is a random vector from F then Y with X is 0 we have shown so X belongs to null space of F V itself is finite dimensional null space of F is also finite dimensional so it has an orthogonal complement so I can write V as null space of F direct some null space of F perpendicular whenever you have a finite dimensional subspace it holds but V itself is finite dimensional so this holds so what this means is that a linear functional on a finite dimensional inner product space is completely determined by its action on null space of F perpendicular F is completely determined by its action on the subspace null space of F perpendicular of course the first term is null space of F so F takes the value 0 there F is completely determined by its action on null space of F perpendicular this is something which does not hold for a general linear transformation for a linear functional this is that is true okay. Using this representation theorem we will show how the conjugate transpose it is called the adjoint operator we will show how the adjoint operator for a linear transformation on a finite dimensional vector space exists and it is unique and also derive some of its properties okay that we will do next time.


Welcome to today's vlog. So today I've got my guest Toby. He's back from school So I've done a haircut on him before this one's gonna be a little bit different little more clean cut Slight disconnection not really too much definitely longer on top But the cool thing about Toby's hair is it's a nice and thick coarse wavy hair So I think you guys will get a lot of techniques from that we get out the texture scissors We create texture on the top of his head so a lot of techniques to come Hope you guys like it make sure you subscribe below if you haven't done that already subscribe to the channel Also, let me know if you have any questions or comments below love hearing positive feedback Anything that you guys want to see coming up and also if you have questions about this technique make sure you post them All right guys, let's get started with the vlog here. We go all right guys So as you can see Toby has not gotten a haircut probably since Christmas break So he waits and waits until he comes back from school so I can give him a cut again, which is pretty cool I've been doing his hair for a really long time So I love having a guest like that. So we start off. I go in I'm using a three and a half blade With my andis supra ZR clipper. So this is one of the newest detachable blade clippers that andis has to offer It's super powerful. I love the detachable blades because of the power But the problem is that they don't have half sizes for the most part as you work through So sometimes you need a little extra help with fading. So I'll use the andis Master series clipper towards the end to get a little bit the tighter fade I'll also use a lot of scissor over comb techniques to also Fade the hair as well. So I like to use a few different tools as I'm working with men's hair I'm not a barber. I don't do a lot of tight fading This is a very standard haircut that I would do on a guy who comes in so I use different tools to create the look So we start off again. I'm using a three and a half blade from andis so That's gonna get a nice medium length for me Then I'm gonna go through I'm using my ys park 209 comb with the triple zero blade And what I like about that is I get really tight to the comb So basically where I'm holding the comb is where I'm gonna cut and I go through and do a lot of clipper over Comb work. I'm a clipper over comb guy if you don't like doing that You could fade this more with a blade or a guard if you wanted to But I like going through and doing a little more freehand work When as I'm fading the hair up. So really we're building up a graduated line with the clipper over comb So the thing I like most about the 209 comb and using that to work through first off It's about the size of the clipper blade. It's a little bit smaller than that But not much so I don't have to worry about falling off of the comb and cutting a hole in the hair And also it's a really flexible comb So as I'm working through if I want to get tighter to the scan and kind of fade out Then I can do that with this comb as well We have it in all different colors on our website and freesaloneducation.com This is the white version.  It just allows you to see the hair coming through the comb better So again working my way up now one thing that you'll notice when I'm working clipper over comb now that I'm on this side I'm always trying to work the hair off away from the face first off You don't want to throw hair into your client's face, but second. It's always good to be Working the hair in the same direction. So if I'm working the hair back off the head on the one side I want to do the same exact thing on the other side. So now this is where I was talking about I'm going through this is using a zero guard With my andis master series clipper So just to tighten it up a little bit more around the hairline give it a more natural hairline This is something I've learned from putting out Quite a few different men's cutting videos and the biggest comment usually is why didn't you take that neckline a little tighter?  We go a little bit tighter gives it a little more longevity Really when I think about why am I doing this? I'm doing it for Toby's sake because he doesn't come in for haircut as often So I go a little bit closer with it and fade it up Now we're gonna go through this is a ys park spray bottle If you guys are interested in getting a spray bottle like this, let me know We're thinking about bringing it on to our shop on free salon education So that's something you think you would like then let me know Still using the ys park 209 comb now going in with my Mizetani This is the solid blacksmith fit solid scissor at a seven inch length The reason I like using seven inch length for scissor over comb is because it gives me Not more hair to cut so I think a lot of people Think of it as I can cut more hair at a time You're not really doing that But it gives me a longer reach to reach over the comb to allow me to have Easier access to that hair that's coming up. So Seven inch Mizetani blacksmith fit solid now if you're looking at getting a new scissor You could check out free salon education comm use the code Matt Beck vlog you get 20% off the thing with the Blacksmith fit solid as it has really small finger holes I like that personally, but if you're somebody with really big fingers, then you might not like that scissor as much Maybe go with a blacksmith fit Regular scissor, but just get a longer length for that scissor over comb So as I work across the top of the head everything's coming straight out. So no over direction for the Basically the entire top of the head then when I get close to the forehead That's what I'm going to start to over direct it back The other thing that I'm doing here is I'm blending it a little bit into the side So you'll see me kind of rounding it off the reason I'm doing that is I don't want it to disconnected He's got really thick hair So if you had fine hair, I'd probably go straight across the top I wouldn't round off the sides of the corners like I am right there But because he has thick wavy hair, he's going to build up a lot quicker than somebody with fine Thinner hair. So now we're going in with the Mizutani type Z dual texture scissor The thing that's cool about this scissor is that there's no blade That gets in the way. So you go through both sides of the blade or texturizers So we go through we just do a little bit of blending work there We're gonna do a little bit more on the top in a minute. This is the Donald Scott naked blade Straight blade basically so I've been working with this a little bit They sent it to me. We're thinking about carrying this on the shop as well Let me know if you're interested in that. So I go through and just do the detail work around the edges All right, the last thing that I'm gonna do is we're gonna go in with our Mizutani type Z dual texture scissor And this is really gonna create texture take out weight a lot of guys with this type of hair I feel like it gets thick and bulky. So just going through doing some point cutting with the type Z dual texture scissor It's a really cool scissor for that and now I'm gonna go through and work on my styling Styling is really important with men's hair I think a lot of guys don't take the time to do it But if you think about it if I put product in my wet hair, it's just gonna dilute the product and you won't get that Hold all day. So we're gonna go in with the adh dry. This is Andrew does hair product. He sent it to me He's a good friend So I wanted to use it on the video So a lot of you guys probably know who he is if you don't follow at Andrew does hair on Instagram You'll see some of his work and you can pick up his product as well But we put the product in his whole motto is good hair doesn't come from a jar. It's all about the style all about the cut really just Creating that look blowing a dry nice and then just adding the product in to get that finish in touch So hope you guys like this cut. Let me know in the comments below what you think and if you have any questions Let me know as well. Thank you guys so much for watching this video Don't forget check out free salon education calm We have over 600 videos also use that code Matt Beck vlog at checkout get 20% off a lot of different products on our web store and Make sure that you join our FSE partner program That is a program where we do live classes for free for stylus All you got to do is sign up go to FSE Partner comm you can sign up for that for free You'll get the alerts about the live classes and also if you are not a stylist and you like this cut and you're looking for a New hair salon go to our website hair salon locator comm That's where we put all of our FSE partner salons so that you guys can find a great stylist to create this look for you Thank you guys so much for watching make sure you subscribe and I'll see you guys on the next video. Thanks


Now, while we do realize that it's still the preseason, and we know teams are putting everything on for display, whether it be on offense, defense, special teams, they're not showing their hand in preseason because these games just simply don't count. Well, don't tell that to the Baltimore Ravens because we got that 24 game winning streak. And I love it. I love it and I respect it. And it makes the preseason, it makes watching the preseason games that much more exciting. But during preseason, even though the games do not count, they do count for players as far as earning them roster spots. So these games are not all for nothing. But sometimes we do like to see potential in some guys. We like to see some guys flash. We like to see some guys put on a show because that can get us that much more excited for when they end up playing in the regular season. And our first question came from my guy Brandon. And he asked the question, I haven't heard anything from Trenton Simpson. Is he hurt? Not too much talk about him. I'm eager to see him perform and fit in. And I think we all are. Trenton Simpson Ravens, third round pick from Clemson. And I remember when they first drafted him and Patrick Queen tweeted, sheesh. I was thinking, uh-oh, yep. I already thought that it was going to be the beginning of the end for Patrick Queen. But when they drafted him, I was like, oh, yeah, that's a wrap. But Patrick Queen is still with the team, which I'm extremely glad for because y'all know me. I want the Ravens to get everybody and to keep everybody. But anyway, we'll talk about that another day. But with Trenton Simpson, I do feel like it's been pretty quiet surrounding Trenton Simpson this offseason. So I just really don't know the status of a Trenton Simpson. Y'all please, team keeping clean, if there's anything that I missed when it comes to Trenton Simpson, please let me know. Because he's somebody just like my guy Brandon said that I'm excited for. I'm excited to see what he does. I'm excited to see how he does and exactly how Mike McDonald uses him. Exactly how Mike McDonald can tap into the potential that the Baltimore Ravens saw when they drafted him in the third round this year. And exactly are they going to have him rushing the pass? Are they going to have him dropping back? Are they going to have him blitzing? Just really doing a little bit of everything. Where they going to have him lined up? It's just, it's fun to think about that. Like it's fun like this, y'all know, too. Well, I've been one, I've been one Ravens to do it on offense for the longest. And this year they finally did it. But it's fun when you have so many different weapons. It's fun to just imagine how they'll all be and how they'll all fit on the field together. We've been one that on offense for the longest. Now we finally got it. But on defense, it's been a thing where they've pretty much been doing it every year. But this year too, this year is no different. Cause you got of course, Roquon Smith and you got Patrick Queen.  You got David Ajabo, you got Trent Simpson now. You got Kyle Hamilton. Right now you don't have Marlon Humphrey or Rocky scene for now. But those guys will be back soon enough. But just to see and wonder how they'll use Trent Simpson is interesting. And it may be the won't truly be something that we won't truly see until week one of the regular season. But until then we'll just wait it out. Next question came from my God, Michael. And he said, this is my first time and he said, hey, team, keep it clean. I've been a long time subscriber for years now and love the platforms. I appreciate that, Mike. Thank you, man. He said, but I'm a Ravens fan that never been to a Ravens game. So I'm planning on going to my first game against the Texan. So I know you've been to a game. What should I expect? I wanted to know what's the difference between watching at home and watching in the stadium.  But you got me hyped for you, man. I'm real happy for you, man. It's your first time going and it's gonna be a home game and it's week one, two, man. Like you, like there's no wrong Ravens game to go to. But you sure picked a really good one because with all the hype and all the excitement that we have and just all the, we're just ready, ready to see these Baltimore Ravens play. And we ready to see all the starters be out there. We ready to see this full team be put on display. You get to watch it live. What should you expect? What's the difference between watching it at home versus watching it at the actual stadium? The energy, because you're right there, like in the middle of everything. And I know, like sometimes my wife would be like, when she's, if I'm looking at a game or something, she'll be like, oh, let's get, she'd be thinking, oh, we should sit in like one of the suites or something. I'm like, no, I want to be out there, man. I want to be out there like with everybody in the crowd because it's just different. Like being in a suite, cool. You can get all the food and stuff and the AC. I don't care about none of that during the game. That's just me. I don't care about none of that though. I want to be out there in the heat in whatever the weather is. If it's cold, I don't like the cold, but still I want to be out there because just hearing the crowd cheer or boo after a great play or a terrible play or whatever it is, it's just different, man. And it's so much fun, man. It's so much, I promise you're going to love it. You're going to love it. I mean, you're going to love it more if the Ravens win, but you will love it, I guarantee you. The big difference between being there versus watching on TV. First and foremost, just the play of entrances. That is one thing. Again, Ravens do such a great job with the fan experience. I always say that, man. So get there early. Please get there early. Be there by 1250. Kickoff is at 1 PM, I believe. So be there by 1250. Don't be there by 1250. Make sure you are in the stadium. I say, I really say by 12. It's your first game. You know what? No, go tailgate. Go tailgate. Just go walk around. Go walk up and down. Ravens walk and whatnot. Go to the different lots. See if you can get there about... What time do players come in? I think they come in at about, the game starts at one. So the players probably start coming in about 1130, I believe. Is it a lot? What lot is that? It's the lot where all the fans line up across the gate. What you could do is get there about 11. And I forgot what lot it is. I know somebody in Team Chemical ain't gonna know that. They can put it in the comment section for us. But the lot where all the players park in and then they walk in. So you can see all the players walking in and stuff. And that's a cool experience. So get there at 11. I think they walk in at about 1130. And if you don't know which lot it is, cause I don't know which lot it is, you can just walk around the stadium and you'll run into it. Cause you'll see fans lined up and you'll see a gate for people to walk in it and you'll be able to see everybody there. So get there at 11, do that. And then you can do some tailgating. You can walk around the stadium and whatnot. Walk around outside the stadium with the different lots and whatnot. Walk up and down Raven's Walk. I know a lot of people like going to see the statue and stuff. Ray Lewis and Unitedson and whatnot. So you go do that. Go do that. Just have that full experience. If this your first game, do everything. Cause it's, it ain't nothing like the promise you, man. You're gonna love it, man. You're really gonna love it. Then get to your seat. I say get to your seat. Try it like by 1230 to be straight.  Of course it's gonna be some Texans fans there. It's gonna be a couple of them. But it's gonna be mostly Ravens fans. It's gonna be cool to just be around that. And then when you, after you enter the stadium and then you enter the section where your seat is in, just walk into your seat and then you start looking around and stuff. It's gonna be like, oh man.  You're gonna love it. You're gonna fall in love before the game even start. I guarantee you. What else? And then, yeah, the player entrances usually start about like 1250, about 10 minutes before the game, 10 minutes before kickoff and whatnot. During the game, big difference.    So that's gonna be your commentary.  I mean, really, any seat is good. You're gonna be able to see it. Every single seat has its good stuff and some had it bad stuff too, but you're gonna be able to see wherever you sit at. So yeah, the big screens, just in case if you have a seat where, say for instance, you're on the visiting end zone, whatever, you're on what you're sitting in the end zone then that means when the offense is, whether Ravens offense or Texans offense, if the offense is in the red zone, you're ready to see really good and you won't have to look at the screen because you'll be sitting in the end zone. But if they're on the opposite end where you sitting at, you might not really be able to see as much, but they got the screens for that.  You're gonna be straight, man.  How long do they put in there? Like three, four years ago, I think? Something like that. But anyway, you're gonna love it, man. You're gonna love it. Anybody else you wanna put anything, any suggestions for them in the comment section to feel free, go edit, but you will love it. I guarantee it. You will love it so much. And by the time, not even before the game even starts, but definitely once the game does start, but before the game even starts, you already gonna be looking up, when the next time Raven's gonna play near me so I could go again? The next question came from my guy, Jarvis. He said, it's been a while since I had a question for you. So here I go with the failed drafting of Hollywood. What? That was not a failure. It wasn't a failure of a drafting at all. So I mean, that's how you start knowing for like, oh, anyway, let's keep going. He said with the failure, excuse me, with the failed drafting of Hollywood, if you could go back to that draft night, who would you had selected instead of Hollywood? Keep it a good work, Mr. Clean. And I know you hype about the season, especially getting to see Clowney as a Raven, LOL. Also, how do you see Hollywood career panning out cause he's been a no-show since he left us? No, he hasn't been. He was actually one of the top receivers in the league, numbers-wise, and then he got hurt. That's what it was. He got hurt last year. That's what it was. He was doing just fine, but the injury got him. And he said, last question, which Raven do you think will have a great season? I say Lamar for sure. The Ravens pass catchers as a whole. Everybody who catches passes because they can really feed off of each other. And I think that Ravens offense can help the Ravens defense have a great season and just make their lives a lot easier. And hopefully they can mask Marlon Humphrey being out. I know that, of course, there gonna be some hiccups here and there, of course, cause you ain't got Marlon Humphrey. So that's a big loss, but to absorb the blow Ravens offense will really have to just take care of them. But really going back, who would I have taken instead of Hollywood? I feel like you phrased that like Hollywood was a bad pick. He wasn't. He wasn't Hollywood. And he did his thing, like literally for the first game with the Ravens. And then even a second game, remember? First game, well, he had them two touchdowns. His first catch was a long touchdown and he had another long touchdown too. So he was killing them, that first game against the, in Miami against the Dolphins. We were there. Man, that might've been probably favorite live game of the, no, well, one of them, I got a couple of favorite live games that we went to from Lamar Jackson. Probably that one for sure. But also the one the year before, his rookie year, being at the week 17. Ooh, that game was crazy, man. Week, that week 17 game was crazy, man. Well, yeah, week 17, because they didn't have 18, I mean, they didn't have 17 games back then. But when he was a rookie, with him versus Baker Mayfield for the division, and everything back, anyway, we getting off subject. But Hollywood, yeah, he showed out that first game. Then the second game too. I remember, we was at that second game too, the 2019 season, because we went to the game in Miami, then we flew out to Baltimore the very next week to go to that game too against the Cardinals. And Hollywood made that clutch catch. And Lamar made that clutch throw at the very end and it was beautiful, man. But Hollywood wasn't a bad pick, man. He was not a bad pick. He came to this thing. Did he have some drops? Yeah, he did.  Yeah, he did.  I would have drafted him. Now, there's some other good receivers in that draft, for sure, for sure. But I don't think it was a failure of the Ravens to draft Hollywood. And the last question on this special episode, a question from subs right before our pre-season game against the Commanders tonight, which will be live streaming. I hope to see y'all there. I came from my guy, Liam. He said, hello, Engraven. Hope you and the family are having a wonderful day. Yes, this Monday is starting off great. Hopefully it finishes off great with the Ravens winning. Even though it is pre-season, but still. He said, this is my first question being sent to Team Keep It Clean. I've been watching you for over three years now. Yeah, back. Hey, hey. He said, I've been watching you for over three years now. Yeah, back when you put yourself in your thumbnails. Hey, man. Cause I just say, no, but and I still like the thumbnails now. There ain't nothing crazy, man. But it's a process, man. It's a process. I even look at thumbnails from like two years ago and just some, I've been looking at them now. But it's a process, man. This whole, this whole thing, as you know, has been a process, but I appreciate you just hanging out for the process and hanging around and whatnot. But yeah, man, the thumbnail part. I used to be so frustrated because I just, I did not know how to make them at all. Like, and again, now they ain't, they ain't nothing crazy. They is super, super, super basic even now, but I just did not know how to. So I just had to take a picture of myself. And that was that. So shout out to anybody that was there through that almost like... I appreciate that, Leigh. I appreciate you bringing that up They said, anyway, back to my original question. What are your predictions for this year's Baltimore Ravens... I can't do this one. What are your predictions for this year's Baltimore Ravens 2023, 53 man roster? Who are some of your dark horses to make it? And who do you have getting cut by the team? Also, who do you think will be this year's MVP? Offensive player of the year, defensive player of the year, comeback player of the year, offensive defense rookie of the year. Let me know what you think. And as always, team, keep it clean. Appreciate that, Leigh. So I can do some predictions for the 53 man roster. I can't do the whole thing. I think Lamar Jackson is going to make it. Now that number two quarterback position there. Hold up there. As of pre-season week one, that's Tyler Huntley's job. But now Tyler Huntley's hurt. I think he'll be straight by next week, but that's just me. I ain't hear nothing. I don't know nothing. Y'all know I'm an NFL outsider. So no sources, no plugs, no nothing. But anyway, as of the first pre-season game is Tyler Huntley's. Would they keep three quarterbacks? They do have that rule where you could have a third one on game day. So I forgot exactly what the rule is. Now I don't know with that rule, if the quarterback, the third quarterback can be brought up from a practice squad to fulfill that rule, or they have to be on active roster, I don't know. If they can be on a practice squad, then I think they'll only keep two. If they can be brought up from a practice squad, then I mean, if they have to be on active roster, then I think they'll keep three. But if they can be brought up from a practice squad, then I think they'll keep two. As far as running backs, right now for the first pre-season game, I think Keith Mitchell makes it. I think Melvin Gordon is actually the odd man out. I think it ends up being J.K. Because it was Justice Hill and Keith Mitchell. Y receivers, we know the top five already. As far as six right now, I think it's Tylen Wallace. In order for Tylen Wallace to be supplanted, somebody to take over for him, somebody would just have to go out there and destroy everything in a good way as a Y receiver. They would have to just kill it. Yeah, so that's really all I got for the 53-man roster because we'll be here forever if we go down every position. Some of my dark horses to make it on the team. Before the pre-season game, it was Shamar Bridges. It was Shamar Bridges, but hey, we got two pre-season games to go. So we'll see. Like he would, again, he would have for him to make it over Tylen Wallace, especially Tylen Wallace being drafted. Well, he's a fourth round pick a couple of years ago. Him being drafted, he got to leg up. Raven's really liked him, so he got to leg up. But Shamar Bridges would really have to just show out like crazy. And Tylen Wallace would be on special teams too. So, again, that's why I feel like Tylen Wallace, not that he got it on lock, but he holding it down right now. So we'll see. Who do I have getting cut by the team? Again, Melvin Gordon, I don't think that he'll make it. As of right now, of course things could change with Melvin Gordon. I could see James Proshay. Because again, it's sad how everything just, how it's been, it just, I don't think it's worked out. And I mean, I don't even really got to go down to all the other receivers. Because I think right now, Tylen Wallace is going to be the one to make it. But I mean, every all the other receivers get cut. So I would like to see some more from Laquan Treadwell too, to see what he can do. But yeah, I think that's Tylen Wallace's spot. Obviously there's still a practice squad for all the receivers and stuff and all these other players and stuff too. And I think a dog horse to possibly get cut, it would be crazy. And I said it before, it would be crazy. Like, it would be Paracord. Yeah, it would be crazy. But I don't know. Again, they trying to mount all these different places. I don't know. So and then I don't know. Again, I don't think he'll be cut. But you said dog horse. That would be a dog, dog horse, like a horse dogged in me. But anyway, he also said, who do you think will be this year's MVP? Lamar, offensive player of the year. Lamar, defensive player of the year. That's a good one. Roquan Smith, I mean, obviously Lamar, I feel like it's an easy answer for those first two. Roquan Smith is an easy answer for the second one, the defensive player of the year. But if I got to choose somebody besides Lamar for MVP and, or for somebody besides Lamar for offensive player of the year, I will go Bateman. I'll go with Shaw Bateman. Bateman, man, I've been talking up all this, all season, man. I know you're gonna come through. I know you're gonna come through and show them why you, I receive a one, uno. You might win number seven, but you, I receive a one. So Bateman, defensive player of the year besides Roquan. I'm gonna say this is his year. This is our guys year to turn the corner. Adafi away, defensive player of the year for the Ravens. And I know, I know that will sound real crazy, but hopefully looking back at this video after the season's over, he'll be like, oh yeah, yeah, yeah, yeah. That was right. Comeback player of the year. Well, I mean Lamar, because he missed the end of last year and now he came back. So, and then offensive and the offensive and defensive Roqui, the offensive, I could go Zay, defensive, just on the Ravens. Well, maybe Trenton Simpson. Like I always talked about at the top of this video. So yeah, we'll see. We'll see if my predictions end up coming out to be true or false. I love y'all team, Keep It Clean. I appreciate y'all. Make sure you subscribe to the channel, leave a like on the video so y'all miss nothing. And I will see you all, even though you know these Ravens, they probably up to something. They probably gonna end up doing something between now and the preseason game. But I'm telling y'all, I'll see y'all for the live stream for the preseason game tonight, but it'll probably end up being before then knowing how these Ravens get down. I love y'all, man. I'll see y'all soon, man. We out.


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