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* = (*v*_{1}, *v*_{2}), then the first element of *AV* is *v*_{1} + 2*v*_{2} and the second element is 2*v*_{1} + 4*v*_{2} = 2(*v*_{1} + 2*v*_{2}). Note that if *v*_{1} + 2*v*_{2} = 0, then *AV* = 0; and so V, V = *V*^{T} *AV* = 0. This shows that , is *not* an inner product since any nonzero vector *V* with *v*_{1} + 2*v*_{2} = 0 satisfies *V*, *V* = 0 (thus violating the positivity axiom).

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