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S) HELP. 2. Calculate R S for each of the following choices of R and S: 3. 56) and express yl, . in terms of . X I , . . (i) by eliminating u l , . . and (ii) by multiplying the coefficient matrices: 3ul + 2uz = yl 5x1 - X 2 = U I and 5 ~ 1 6ul = y2 . rz - x s = ul 2x1 3x2 X 3 = ~ + + 2 ' 25 26 Advanced Calculus, Fifth Edition 4. 8: a ) Rule 12. b) Rule 14. C ) Rule 18, by induction with respect to I . d ) Rule 19, by induction with respect to 1 and Rule 18. 5. Let A be a square matrix. Prove: a ) A' - I = ( A + I ) ( A - I ) .
H, are the (necessarily real) eigenvalues of C (with multiple eigenvalues repeated in accordance with their multiplicities). For a proof of this theorem, see Section 5-3 of the book by Perlis listed at the end of the chapter (see also Problems 12 and 16 following this section). This result is very important for the study of quadratic forms. Let C be a symmetric matrix and let A be an orthogonal matrix such that A-'CA = B = diag ( A l , . . , A,). Then let us consider the quadratic form . We express x l , .
Vk are linearly independent and h < k, then vl , . . , vh are linearly independent. e) (Rule for comparing coefficients). If vl , . . , vk are linearly independent and then a1 = bl, a;, = b2,. . ,ak = bk. f ) There exist n linearly independent vectors in Vn: for example, the vectors (4 g) There do not exist n + 1 linearly independent vectors in Vn. h) If v l , . . , vn are linearly independent vectors in Vn, then v l , . . ,v, form a basis for Vn; in particular, e l , . . , e,, form a basis for Vn.