Find the matrix M of the linear transformation T : R3 → R2 given by () [ = M: T -9x1 + x2 + (-4)x3 -721 +3x2 (51)23] 9. 지

Let T: R² R² be defined by T -{[5].[3]} C = Given Pc = [ [T](PB (u)) 1 2 = 3x1x2 -3 ([2₂])-[¹22] Lu= [2] B-{].G]}, and [3]} = Let -x1 + Ex: 5 x1 = -2 , use the Fundamental Theorem of Matrix Representations to find [T](PB(u)). 5

Find the standard matrix for the linear transformation T(1, 9) — (2а — Зу, т — у, у — 4г) -3 1 -1 1 1 -4 2 1 -4 -3 -1 1 -3 -1 2 1 1 -3 -1 2.

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 =

[:] Suppose that T is a linear transformation from ³ into ² such that 3 -[-2]. (₂) - [8].a T -[2], and T(3) = [1] The columns of k T(e₁) O *1 Tx2 = 2x1 x3 *1 TX2 002 1 0 0 0 Tx2 = *3 3x1-2x2] Tx2 = 2x1 001 Find a formula for the image of an arbitrary x= x2 in ³. 4x2 + x3 1 0 are e₁ 3x1 + 2x2 - 4x3 -2x1 + x3 3x1-2x1 2x1 3x1 + 2x2-4x3 -2x1 wwww + X3 x3 0 e2 ез 0 akan wwwwww.

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4a). The transformation defined by the matrix A = [14 5 stretches images in R² in the directions y = x and y = -x. Figure out the factor by which anything in the y = x direction is stretched and the factor by which anything in the y = -x direction is stretched. 4b). The transformation defined by the matrix B = -82 -55 13 3] stretches images in R2 in one direction by a factor of 3 and some other direction by a factor of 2. Figure out what direction gets stretched by a factor of 3 and what direction gets stretched by a factor of 2. 4c). The transformation defined by the matrix C = stretches images in R2 in two directions. Find the directions and the factors by which it stretches in those directions.

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-9 V₁ = --[-] and X₂ such that T(x) is Ax for each x. Let x = A = X₁ -2 [3] and let T: R² R² be a linear transformation that maps x into x₁v₁ +X₂V₂. Find a matrix A -8 and V₂ =

For a 4×4 matrix whose top three rows are arbitrary and whose bottom row is (0, 0, 0, 1), show that the points (x, y, z, 1) and (hx, hy, hz, h) transform to the same point after homogenization.