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MATH 2270-2 FINAL EXAM-F11 NAM E Do not write below this line! 1. (15) 2. (15) 3. (10) 4. (10) 5. (10) 6 (30) 7. (30) 8. (30) 9. (20) Total 170 2 1. Let A ∈ Mm×n . Answer questions a, b, c (below) for this A. (5) a. Can the rank of A be k where m < k < n? Explain. (5) b. If n < m and rk(A) = n does AX = Y always have a solution for all Y ∈ Rm ? Explain. (5) c. If m < n does AX = 0 have a unique solution? Exlpain. 1 2. Let A be row equivalent to 0 0 2 0 0 (10) a. Find a basis for the null space of A. (5) c. Find a basis for the row space of A. 1 1 0 0 0 . Solve a, and b (below) for this A. 0 3 (10) 3. If {(1, −1, 1), (0, 1, 1)} is a basis for N(AT ) where A ∈ M3×2 then find a basis for Col(A). 3 (10) 4. Let L be the thru the origin and having direction vector √ line in R passing 3 (1/2, 1/2, 1/ 2), and T : R 7→ R3 be the linear transformation T (X) = X−ProjL (X). What is the image (= {Y ∈ R3 | Y = T (X) for some X ∈ R3 }) of T ? Draw a picture! 4 1 −1 0 (10) 5. Let A = . Show that σ = 1 is a singular value of A by finding 0 1 −1 V ∈ R3 , U ∈ R2 of length one such that AV = U. 5 6. Let B = {X1 , X2 , X3 } be a basis for R3 where X1 = (1, 1, 1), X2 = (0, 1, 1), and X3 = (0, 0, 1). Solve a, b, and c (below) for this basis. (10) a. Find the coordinates of (3, 1, 7) relative to the basis B. 1 (10) b. If [A]B = 0 1 −1 0 0 1 and [X]B = (1, 1, 1) then find AX. 1 1 (10) c. If T (X) = (x1 + x2 , x2 + x3 , x1 + x3 ) with X = (x1 , x2 , x3 ) then find [T ]B . 6 7. The Gram-Schmidt process applied to the vectors X1 = (1, 1, 1, 1) and X2 = 1 1 (1, 9, 9, 1) gives an orthonormal pair U1 = (1, 1, 1, 1), U2 = (−1, 1, 1, −1). 2 2 Solve a, b, c (below) for these vectors. | | (10) a. Find the QR factorization for the matrix A = X1 X2 | | (10) b. Find the matrix of ProjV where V =span(X1 , X2 ). x (10) c. Find the least squares solution to AX = Y where A is as in part (a), X = , y 1 2 and Y = . 3 4 7 1 8. Let A = 0 1 0 1 1 −1 1 . Solve a, b, and c (below) for this matrix. 1 (10) a. Find the characteristic polynomial for the matrix A. (10) b. Find a basis for the eigenspace R3A (1). (10) c. Is A diagonalizable? Explain. 8 (10) 9. Determine if the quadratic form q(x, y) = 6x2 + 4xy + 3y 2 is positive definite. (10) a. Draw the curve q(x, y) = 1 where q(x, y) is as above, giving the coordinates of vectors which lie along the principal axes.