{x – x², 2x – x², – 1+x}, {-1-г+2?, 1 + 2х — а?, 2т — г?}. B C a. Find the change of basis matrix from the basis B to the basis C. [id b. Find the change of basis matrix from the basis C to the basis B. [id =

2. Let P denote the vector space of all polynomials on R. Show that P is infinite- dimensional.

Let S {V1, V2, V3} and T = {w₁, W2, W3} be two bases in the vector space P₂ {a+bx+cx²: a, b, c = R}, where V₁ = 1+x+3x², v₂ : −2 − 2x − 7x², V3 = 1 + 2x + 6x² and = : 1 + x − x², w₂ = W1 = 1+ 2x², W3 = 3-x+5x2 Find the transition matrix from T to S. Let v = 201 - 3v2 + V3, find the coefficients (C₁, C2, C3) so that v = C₁w₁ + C₂W2 + C3W3. Hint: For the second part, you need also to find the transition matrix from S to T. =

Consider the basis {1, 1 - t, 1 + t + t2} of the space P2 of polynomials of degree at most two. Find the coordinate vector of t - t2 with respect to the standard basis.

Let V be an n-dimensional vector space over the field of complex numbers C. Showthat V over R has dimension 2n.

Let A X = = −1 1 -1 −1 −1 1 -1 -1 -1 −1 1 1] 1. Find a non-zero vector x in the null space of A. −1 1