By Sam Perlis
The major subject matter of the e-book is the institution of the well-
known canonical varieties. Rank, nonsingularity, and inverses are
introduced in reference to the improvement of canonical matrices
under the relation of equivalence, and with no the intervention
of determinants. Congruence and Hermitian congruence are the
next significant subject matters.
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Example text
This covers all the possibilities and completes the proof. C1 EXERCISES 1. Solve each of the following systems by reducing the augmented matrix of each system to the simplified form of Theorem 3-3. (a) + x, = 3, 2x, + x, + 2x, = 3, { 3x, + 2x, - x, = 0. (b) X2+ X3==3, 2x, + x, + 2x, = 3, X1 + 2x2 X3 = 0. x, - x, Xi- { (c) 2x,+x,+ x,=8, { x, + x, + 3x, = 10. (d) { + X2 + = 0, x, + x, + 3x, = 0. 2x1 X3 2. Without finding any solutions, show how to prove that the system (d) in Exercise 1 must have a nontrivial solution.
T, in a vector space V are linearly independent and span V, they are said to form a basis of V. (ir) is readily found: u, = (1, 0, ... , 0), u. = (0, I, O, ... , 0), u. = (O, 0, ... , 0, I). These are obviously independent, and every vector t = (x,, ... ) is a linear combination of them: ~ = X1U1 + ••' + XnUn, These vectors u1, ... , Un are called unit vectors, and the notations Ui will be reserved exclusively for them. There are, however, many other bases - for example, u,, ... _,, and t = (I, I, ...
F, they are called linearly dependent if there are scalars ki, ... , k,, not all zero, such that (9) If no such scalars exist, the set /;1, ••• , I;,, is called linearly independent (briefly, independent). The vectors I;, = (1, 2, 1), I;, = (0, 1, O), 1;3 = (2, 0, 2) 28 [CHAP. 2 VECTOR SPACES in V,((ll) are linearly dependent, since 2~, - 4~, - ~. = 0. But if then h~, + k~, ~ (2k, h, 2k) = (0, O, O), so that h = k = 0. Thus ~2 and is a linear combination of them: ta are linearly independent, while ~1 t1 = 2t, + tt,.