The following questions are from a conceptual math class:

2 4 2 4. The matrix A = 0 3 2 has an eigenvector -2 . What is its corresponding -4 -3 4 eigenvalue?2 4 2 4. The matrix A = 0 3 2 has an eigenvector -2 . What is its corresponding -4 -3 4 eigenvalue?7. Compute the eigenvalues of the linear function X 4X 5Y \"(i/l) (2X -2y) 9 8 12 11] Is 3. What IS a corresponding eigen 8. One of the eigenvalues of the matrix [ vector for it? 4 5 3 9. One of the eigenvalues of the matrix 4 6 4 is 2. What is a corresponding 8 11 7 eigenvector for it? 10. a) Solve for a and b in the equation 6(3) + bl?) = (194) b) Use your answer to part (a) to give the coordinates of ( basis (3), (13). 11. Give the coordinates of ( 9 14) With respect to the 7 5 > with respect to the basis (1), (1.2). 1 12. The point of this problem is to demonstrate that if you know all the eigenvalues and eigenvectors of a linear function f (or a matrix M), you can compute f(W) (which is MW) for every vector W. In short, knowing all the eigenvalues and eigenvectors is equivalent to knowing the function. a) Solve for u and v in the equation \"(3) + \"(13) = (194) (Hint: You will probably want to rewrite this as a system of equations and "solve simultaneous/y. ) b) Explain what your answer to part (a) means about the coordinates of (194) in some nonstandard coordinate system. (Hint: Which one?) c) Suppose that f : R2 > R2 is a linear function and its eigenvectors are as follows: (E) with eigenvalue 2, and (13) with eigenvalue 3 What is 11(3))? What is 11(3))? 9 14 and (c) and the two defining properties of a linear function.) d) Continuing from part (c), what is f(( ))? (Hint: Use your answers to parts (a) 13. Diagonalize the following matrices: a) [180 :31