7. 8. Show that T(x1, x2, x3) = (3x₁ − 6x3, x2 + x3, X1 X2 - 2x3) is a linear trans- formation by finding a matrix that implements the mapping. Note that x1, x2, and x3 are not vectors but are entries in vectors. Determine if the linear transformation from the previous problem is (a) onto (b) one-to-one

2. The Stretching Problem Imagine a linear transformation T: R² →R² that has the following properties: In the direction along the line y = -3x, the transformation stretches all points by a factor of two. In the direction along the line y = x, the transformation keeps all points fixed. 2a).Use the space on the right to sketch what should happen to the image shown on the left when it is stretched according to the transformation described above. You may use a combination of intuition or calculations, as well as any additional sketches below or on your group's whiteboard. IN

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.

5. Determine the matrix of the transformation that first reflects points through the horizontal x-axis and then reflects points through the line y = x.