1. Let A be an n × n matrix and let the transformation T be defined by T(x) = Ax. (a) What is the domain and codomain of T(x)? (b) Show that T() is a linear transformation.

Let T:P2P4 be the linear transformation T(p)=x2p. Find the matrix for T relative to the bases B={1,x,x2} and B={1,x,x2,x3,x4}.

Let f be the linear transformation represented by the matrix A = 4-(32) Find the point (x, y) such that f(x, y) = (-4,-1). (Give your answer exactly, not as a decimal approximation.) The point is

Let T1 and T2 be linear transformations given by X1 ЗХ1 + 5x2 X2 -2x1 + 8x2 X1 -2x1 + 9x2 X2 5x2 Find the matrix A such that the following is true. (a) T1(T2(x)) = Ax A = (b) T2(T1(x)) = Ax A = (c) T1(T1(x)) = Ax A = (d) T2(T2(x)) = = Ax A =

(3) Consider the linear map T : R² → R² which sends (1,0) → (-3, 5) and (0, 1) → (4, – 1). (a) What is the matrix of the transformation? What is the change of coordinates matrix? Do they agree? How come?

Let Find the matrix C of the linear transformation T(x) = B(A(x)). C = 4-[]-[8] and B A 5 3

Q2. Let T: R³ R³ be a transformation defined by - ()-() T (a) Is T a linear transformation? Justify your answer! (b) If T is a linear transformation, find the matrix of T.

Let T be a linear transformation from P₂ into P₂ such that T(1) = x, T(x) = 1 + x, and 7(x2) = 1 + x + x². Find 7(1 - 7x + 2x²). T(1 - 7x + 2x²) = Need Help? Read It Need Help? Let Dx be the linear transformation from C'[a, b] into C[a, b]. Find the preimage of the function. (Use C for the constant of integration.) Dx(f) = 8x + 6 Watch It Read It

B) Find the standard matrix for the transformation T:R² R2 defined by the formula, then, use it to compute T(x), where x = (1, 4). SHOW WORK, STEP BY STEP T(x₁, x₂) = (2x1-X2, X1+x2)

Let S be a linear transformation from R³ to R² with associated matrix Let T be a linear transformation from R² to R² with associated matrix Determine the matrix C of the composition To S. C = -3 4- [3 2 2] A = -3 -3 2 B= [31]