APPENDIX B

SOLUTIONS TO SELECTED EXERCISES

Chapter 0

3. Suppose A is the given matrix with first row 1 2 and second row 2 4; if V = (v1, v2), then the first element of AV is v1 + 2v2 and the second element is 2v1 + 4v2 = 2(v1 + 2v2). Note that if v1 + 2v2 = 0, then AV = 0; and so bapp02ie001V, Vbapp02ie002 = VT AV = 0. This shows that bapp02ie001, bapp02ie002 is not an inner product since any nonzero vector V with v1 + 2v2 = 0 satisfies bapp02ie001V, Vbapp02ie002 = 0 (thus violating the positivity axiom).

4. (2nd part) Part a: We wish to show that the L2 inner product on the interval [a, b] satisfies the positivity axiom; this means we must show that if bapp02ie003 dt, then f(t) = 0 is zero for all a < t < b. Part b: Suppose by contradiction that there is a t0 with f(t0) ≠ 0. Let = |f(t0|/2 > 0 in the definition of continuity; which states that there is a δ > 0 such ...

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