Show all work to receive full credit. Clearly indicate your answers. Erase or cross out all work you do not want considered. Each problem is worth 4 points.

1. Find the standard matrix of the linear transformation T(x, y, z) = (x 2y + z, y 2z, x + 3z).

2. Let T be the reflection through the xz-coordinate plane in R3 : T(x, y, z) = (x, y, z).

(a) Write the standard matrix A for the transformation T.

(b) Use A to find the image of v = (1, 2, 2), and sketch both v and its image T(v).

3. Suppose T1 : R2 R2 is given by T1(x, y) = (x+ 2y, 3xy) and T2 : R2 R2 is given by T2(x, y) = (y, x+y). Find the standard matrices A1 and A2 for T1 and T2, respectively, and use them to find the standard matrix A for the linear transformation T = T2 T1.

4. The linear transformation T corresponding to counterclockwise rotation in the plane through an angle has standard matrix cos sin sin cos . Use the appropriate matrices of this type to show that the composition of the transformations corresponding to counterclockwise rotations by 30 and 60 is equal to the transformation corresponding to counterclockwise rotation by 90 .

5. The linear transformation T : R3 R3 is given by T(x, y, z) = (x + 2z, z y, x + y) is invertible. Find the standard matrix A for T and use it to find the standard matrix for T 1 .

6. For the linear transformation T : R2 R2 , given by T(x, y) = (x + 2y, x y), find the matrix A for T with respect to the basis B = {(1, 1),(1, 1)} of R2 .