Let V, W, Z be vector spaces. A function that takes any pair of vectors v ∈ V and w ∈ W to a vector z = B(v, w) ∈ Z is called bilinear if, for each fixed w, it is a linear function of v, so B(cv + d, w) = c B(v, w) + d B(, w), and, for each fixed v, it is a linear function of w, so B(v, cw + d) = c B(v, w) + d B(v, ). Thus, B: V × W → Z defines a function on the Cartesian product space V × W, as defined in Exercise 2.1.13.
(a) Show that B(v, w) = v1w1 - 2v2w2 is a bilinear function from R2 × R2 to R.
(b) Show that B(v, w) = 2v1w2 - 3v2w3 is a bilinear function from R2 × R3 to R.
(c) Show that if V is an inner product space, then B(v, w) = (v, w) defines a bilinear function B: V × V → R.
(d) Show that if A is any m → n matrix, then B(v, w) = vTAw defines a bilinear function B: Rm × Rn R.
(e) Show that every bilinear function B: Rm × Rn → R arises in this way.
(f) Show that the vector-valued function B: Rm × Rn → Rk defines a bilinear function if and only if each of its entries is a bilinear function B: Rm × Rn → R.
(g) True or false: A bilinear function B: V × W → Z defines a linear function on the Cartesian product space.