Q1 17 Points Let T: M2x2(R) P2(R), H (2a +b)x2 + (6 – c)x +(c – 3d). Let B = (16 0) (0 :), (1 o) 9)) = (6 7')(*: -) ) 6 :-)) B' = C = (x2,æ, 1) C'= (x + 2, x + 3, x2 – 2x – 6). You may assume that all of the above are bases for the corresponding vector spaces. Q1.1 2 Points Show that T is linear. Q1.2 9 Points Compute [T),...
Let a T: M2x2(R) + P2(R), 6 d H (2a +b)x2 + (6 – c)x +(c – 3d). с Let B = 9 (6 8), (8 5), (1 3), ( )) (CO 11),( ( 1),66 1 B' 1 1 ? :-)) C = (x²,2,1) C' = (x + 2,2 +3,22 – 2x – 6). Is T invertible? (1pt) O Yes O No
Let a T: M2x2(R) + P2(R), 6 d H (2a +b)x2 + (6 – c)x +(c – 3d). с Let B = 9 (6 8), (8 5), (1 3), ( )) (CO 11),( ( 1),66 1 B' 1 1 ? :-)) C = (x²,2,1) C' = (x + 2,2 +3,22 – 2x – 6). 3 Let A 14). Compute [AB (2pt) Enter your answer here and T(A) C (2pt). Enter your answer here
(11) Let the linear transformation T : M2x2(R) + P2 (R) be defined by T (+ 4) = a +d+(6–c)n +(a–b+c+d)a? (1-1) (i) (3 marks) Find a basis for the T-cyclic subspace generated by (ii) (3 marks) Determine rank(T).
4. (10 points) Let T:P2 + M2x2(R) be the linear map Tax+bæ + c) =( and 1/1 0 1 B' = 0 1 0 0 O 0) (10) be the standard basis for the codom(T). Using 1 any basis B for P2, compute TB.
(9 marks) Consider L: M2x2(R) + P2 (R) defined by a L b d = a + (b + c)2 + dx?. (a) Show that L is a linear transformation, that is, show that L(sA+B) = 8L(A) + tL(B) for any A, B E M2x2(R) and any st ER. (b) Consider p(x) = ao+a1x + a2x2 € P2(R). Show that L is onto by showing that L(A) = P(x) for some matrix A € M2x2(R). Note that you must give...
Let V P2(R) and let T V-V be a linear transformation defined by T(p)-q, where (x)(r p (r Let B = {x, 1 + x2, 2x-1} be a basis of V. Compute [TIB,B, and deduce if it is eigenvectors basis of
5. Let T: P2(R) R3 be a linear transformation such that T(1) = (-1,2, -3), T(1 + 3x) = (4,-5,6), and T(1 + x²) = (-7,8,-9). a. Show that {1,1 + 3x ,1 + x2} is a basis for P(R) (7pts) b. Compute T(-1+ 4x + 2x²). (3pts)
Consider a subset alpha={x+x2,1+x2,1 2x+2x2}ofP2(R). (a) Show that alpha is a basis for P2(R). (b) For f(x) = 1 + x + x2 2 P2(R), find its coordinator vector [f] alpha with respect to alpha. (c) Let = {1, x, x2} be the standard basis for P2(R), and let f(x) = a + bx + cx2 and g(x) = p+qx+rx2 be the elements of P2(R) and k 2 R. Prove that [f+g] = [f] +[g] and [kf] = k[f] and...
linear algebra Let T: P2 - P4 be the linear transformation T() = 2x2p. Find the matrix A for T relative to the bases B = {1, x,x?) and B' = {1, x,x2, x3, x4} A=