Suppose h : P1(R) → R2 is a linear transformation with the following matrixrepresentation with respect to the bases B = {1+x, x} and D ={( 11),( 1−1)}:RepB,D(h) =[ 2 14 2].Find the image of the polynomial 2x − 1 under h.

Find the matrix representation of the derivative map ddx : P3(R) → P3(R), withrespect to the basis {1, x, x2, x3}

Find the matrix representation of the linear transformation T : R3 → R2 givenby the mapxyz 7 →( x + yx + z), where B =111 ,110 ,100and D ={( 10),( 02)}are bases for R3 and R2 respectively

Let Pn(R) = The vector space of polynomials with real coefficients and degree lessor equal to n. Show that the set {x + 1, x2 + x − 1, x2 − x + 1} is a basis for P2(R).Hence, determine the coordinates of the following elements: 2x−1, 1+x2, x2 +5x−1with respect to the above basis

Consider the following.x = 2 0−2, = 1 00, 0 10, 0 01, = 1 11, 0 11, 0 01 in ℝ3(a)Find the coordinate vectors [x]  and [x]  of x with respect to the bases and , respectively.[x]  =  [x]  =  (b)Find the change-of-basis matrix P  ← from to .P  ←  =  (c)Use your answer to part (b) to compute [x] , and compare your answer with the one found in part (a).[x]  =  (d)Find the change-of-basis matrix P  ← from to .P  ←  =  (e)Use your answers to parts (c) and (d) to compute [x] , and compare your answer with the one found in part (a).[x]  =

T: P₂→ P₂ is a linearoperator. Determine whether the set\{T(1), T(x), T(x ^ 2)\} is a basis for P2. 19. . T(a * x ^ 2 + bx + c) = (a + b + c) * x ^ 2 + (a + b) * x + a