In RÂ­3, the standard one consists of the vectors 1Â­0Â­0, 0Â­1Â­0, and 0Â­0Â­1.
The size of this set for the upper field over the lower field determines the degree of any field extension, whether it is Galois or not. The construction of this set is crucial for the Lagrange interpolation formula. These sets can be interchanged stepÂ­byÂ­step with others of the same size, according to the most famous Replacement Theorem. In a common procedure, an element of this set is subtracted from its projection onto the previouslyÂ­constructed elements, and then divided by the norm. A matrix M is transformed to a similar matrix, its conjugate pÂ­MÂ­pÂ­inverse, under a (*) change in this set. One of these sets can be orthonormalized with the GramÂ–Schmidt process. The coordinates of a vector are multiples of these vectors. The size of one of these sets is the dimension of its vector space. In RÂ­3, the standard one consists of the vectors 1Â­0Â­0, 0Â­1Â­0, and 0Â­0Â­1. For 10 points, name these sets of linearly independent vectors that span a vector space.
The size of this set for the upper field over the lower field determines the degree of any field extension, whether it is Galois or not.
basis [or bases; or basis vectors]