ord 1·Consider the linear transformation T : R4 R3 defined by T(xi, r2, r3, A) (
ID: 3114043 • Letter: O
Question
ord 1·Consider the linear transformation T : R4 R3 defined by T(xi, r2, r3, A) (21-32 4,1 +33 +4r4, 72 2r3+3x4) Find the matrix representation for T using the standard bases for R4 and R3 Consider the linear transformation T : P2 Pi defined by T(p(z)) = p(1) + p(2)x where p(1) and p(2) are the values of the polynomial p(x) when x = 1 and x = 2 respectively. (a) Find T(p(x)) when p(x) = 1 +x2. (b) Find the matrix representation for T using the standard bases for Pa (5 (1,) and (c) Let p(z) 2-r + 3r2. Write down [p(x)) , and then calculate [T(p(x)le using your matrix representation from (b). Hence find T(p(x)). Please turn over hematics and StotiaExplanation / Answer
4. We have T(x1,x2,x3,x4) = (2x1-3x2-x4, x1+3x3+4x4, x2+2x2+3x4). Therefore, T(e1) = T(1,0,0,0) = (2,1,0); T(e2) = T(0,1,0,0) = (-3,0,2); T(e3) = T(0,0,1,0) = (0,3,0) and T(e4) = T(0,0,1,0) = (-1,4,3). Now, we know that the standard matrix of T has columns which are images, under T, of e1,e2,e3 and e4. Hence, the standard matrix of T is A =
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