For the linear transformation Tx y z 2x y z l Determine t

For the linear transformation T(x, y, z) = (2x - y + z, l): Determine the image of (3, 7, 1). Determine the pre-image of (5, 1). Let T: R^2 rightarrow R^2 be the linear transformation that first reflects over the x-axis, then stretches the vector horizontally by a factor of 3, and finally, rotates the vector counterclockwise about the origin by 90*. Construct the standard matrix for T. Write a formula for T that does not contain matrices. Calculate T(12, -5). Consider the linear transformation T: R^3 rightarrow R^3 defined by T(x, y, z) = (x + 2y + 3z, y + 4z, 5x + 6y). Construct the standard matrix for T. Write a formula for T that does not contain matrices. Given the bases B = {(1, 0, 1), (0, 1, 2), (3, -1, 0)} and B\' = {(0, -1, 3), (1, 1, 0), (-2, 0, 1)} construct the matrix of T relative to these bases.

Solution

8)

a)
image will be (2*3-7+1 ,1 ) = (0,1)

b)
there can be many preimage possible
one preimage can be (0,0,5)

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