Q. Let L:M24 → M, be the linear transformation defined 3*2 2*2 -1 3 |A, for A in M, | L(A) 15 3*2 · (a) Find the dimension of ker L. (b) Find the dimension of range L.

Select all of the linear transformations from R³ to R³ that are invertible. There may be more than one correct answer. OA. Dilation by a factor of 5 B. Identity transformation (i.e. T(~) = C. Projection onto the x-axis D. Projection onto the xz-plane E. Rotation about the y-axis F. Reflection in the xy-plane for all

38. a. Consider a linear transformation T from R³ to R. What are the possible values of dim(ker T)? Explain. b. Consider a linear transformation T from R4 to R'. What are the possible values of dim(imT)? Explain.

For each question below, T: R" → R" where n = 2 or 3 such that T(x) = y. Which of the following transformations are linear? UA. B. C. OD. E. OF. Y1 Y2 Y1 Y2 Y1 Y2 Y1 Y2 Y3 Y1 = = = = Y2 Y1 Y2 = = 0 5x2 7x1 9 2x1 + x2 -X1 -3x1 4x1 6x1 x1 +8 X2 0 X1X2

y = 2v to evaluate the integral 1-√²-1² I = !! D where D is the triangular region with the vertices (0, 0), (1, 1), and (2,0). Use the transformation x + y = 2u, x- x² - y² dA