1.2 Find a basis B for the domain of I Suh that the mahix for ī alative to Bis iagend. Hogand. T: R²¬R: T(x ]) = (22 + 2y, x+j) %3D 行 B:

Please type the answer to the question and it should not be handwritten.

1.15. Show the following vector identities by writing each vector in terms of Cartesian unit vectors and showing that each component of the LHS is equal to the corresponding component of the RHS. (a) a (b x c) = c (a x b) = b (c x a). (b) a x (b x c) = b(a·c) - c(a b), this is called the bac cab rule. (c) (a x b) · (c x d) = (a · c)(b · d) – (a·d)(b· c). (d) (a x b) × (c x d) = b[a · (c x d)] – a[b (c x d)]. (e) (a x b) × (c x d) = c[a · (b x d)] – d[a (b x c)]. (f) (a x b) · (a x b) = |al2|b|? – (a - b)². %3D 1.21. A at the co %3D (a) Find at (0,0, 2 b) Sho %3D

For each of the following bilinear forms b : V × V → R, state whether b is an inner product and explain why. b(A, B) = tr(AB), where V = M2 (IR) and A, B EV. b(p, q) = p(0)q(0) + p'(0)q'(0) + p" (0)q"(0), where V = P2(R) and p, q € V.