SOLUTIONS TO SELECTED EXERCISES
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 V, V = VT AV = 0. This shows that , is not an inner product since any nonzero vector V with v1 + 2v2 = 0 satisfies V, V = 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 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 ...