By Wilfred Kaplan
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En. Every orthogonal system of n vectors forms a basis for Vn,by the preceding Rule (h); we call such a system an orthogonal basis (or, for a system of unit vectors, an orthonormal basis). Given k linearly independent vectors vl , . . , vk, it is always possible to construct an orthogonal system of k vectors ul , . . , uk,each of which is a linear combination of v l , . . , vk (Gram-Schmidt orthogonalization process). 1 1 l), ul = vl, uz is a linear combination of vl and v2, and u3 is a linear combination of v l , v2, v3; also, vl = ul , v2 is a linear combination of ul and u2,and v3 is a linear combination of u l ,u2, and uj.
Accordingly, to state that v l , . . , vk are linearly independent is the same as to state that A c = 0 is satisfied only for c = 0. In particular, for k = n , A is a square matrix, and A c = 0 is equivalent to n homogeneous linear equations in n unknowns; thus the column vectors v l , . . , v, of A are linearly independent precisely when these equations have only the trivial solution c l = 0, . . , c, = 0-that is, when det A # 0, or A is nonsingular. Accordingly, n vectors v i , . . , v, of V nare linearly independent ifand only $A is nonsingular, where A is the matrix whose column vectors are vl , .
For typographical reasons we sometimes denote this matrix by col ( a l l ,. . , a m l )or even by ( a l l ,. . , a m l ) ,if the context makes clear that a column vector is intended. We also denote column vectors by boldface letters: u, v, . . 49) has the column vectors vl = co1(1,4) and v2 = col(2, 3). We denote by 0 the row vector or column vector ( 0 , . . , 0 ) . The context will make clear whether 0 is a row vector or a column vector and the number of entries. The vectors occurring here can be interpreted geometrically as vectors in k-dimensional space, for appropriate k.
Advanced Calculus by Wilfred Kaplan