Let T : R2[x] → R2[x] be given by T(a+bx+cx2 ) = a + b(x+ 1) + b(x + 1)2 . Let v = x + 1. Find a basis for

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. =

3. Let F be a subfield of complex numbers. Let V be the vector space (over F), consisting of all polynomials, i.e. V = F[r]. Let T:V → V be the function defined by T: f(x) + (x + 1)f'(x). d For example T(x² + x) = (x + 1) . — (x² + x) = (x + 1)(2x + 1) = 2x² + 3x +1. dr (a) Is T a linear transformation? Give proper reasons. (b) Is T invertible? Give proper reasons.

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Let A = 4 1 -2 -7 V -6 1, b = [77], and c = [] Select true or false for each statement. 1. The vector c is in the range of T 2. The vector b is in the kernel of T Define T(x) = Ax.