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Purdue - MATH 255 - Study Guide - Final

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Final review1 1. Consider[2 r 1 −1 ] [ x1 x2 ] = [2 s ] Determine r, s so that the system is consistent; is inconsistent; has a unique solution; has infinitely many solutions. Augmented matrix: [2 r 2 1 −1 s ] → [1 −1 s 2 r 2 ] → [1 −1 s 0 r + 2 2 − 2s ] 2. If[aIf you want to learn more check out ace 161 uiuc

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+ b c + d c − d a − b find a, b, c, d. 3. Show that ] = [4 6 10 2 ] , A. If any 2 × 2 matrices A, B are symmetric, then AB − BA can never be I2. B. For any n × n matrices A, B, T race(ATA) ≥ 0. 4. Find two different 2 × 2 matrices so that A2 = I2; Find two different 2 × 2 matrices so that A2 = O. 5. For any square matrix A, show A − ATis skew-symmetric. 6. Let A, B be n × n symmetric matrices, show that AB is symmetric iff AB = BA. 7. Find a row/col echelon form of −1 2 2 −1 2 −2 8. Solve 1 2 3 1 1 3 0 1 1 0 2 1 x = 8 7 3 . Reduced row echelon form of the augmented matrix: 1 0 0 1 1 0 1 0 0 2 0 0 1 0 1 1The summary and problems do not necessarily or accurately reflect the contents or depth of the exam. The final exam will consist of multiple choice problems. 19. Assume r ̸= 2. Then the inverse of [1 2 r 4 ] is equal to A. [1 2 r 4 ] B. [−2 1 r 2 −12 [2 1 ] ] C. 1 r−2 r 2 1 2 D. 1 r−2 E. 1 r−2 10. Compute det [−2 1 r 2 −12 [−2r2 1 −12 ] ] a11 a12 a13 a21 a22 a23 a31 a32 a33 . 6 terms of (±)a1−a2−a3−. 6 permutations, even: 123, 231, 312, odd: 132, 213, 321. det(A) = a11a22a33 + a12a23a31 + a13a21a32 − a11a23a32 − a12a21a33 − a13a22a31. det(A) = − det(Ari↔rj) (i ̸= j) det(A) = 1k(Akri→ri) (k ̸= 0) det(A) = det(Akri+rj↔rj) (i ̸= j) 11. Compute 1 2 −3 4 −4 2 1 3 1 0 0 −3 2 0 −2 3 = 8 12. Compute |A| = Find adj(A) and A−1. 0 0 1 0 2 0 3 0 0 = −6 A−1 = 2 0 0 13 0120 1 0 0 13. Let A and B be 3 × 3 matrices, and det (A) = 2, det (B) = 3. Then det (AB−1) = A. 6 B. 2 C. 3 D. 23 E. 32 14. Let A = 2 a 3 5 b −1 1 c 1 , b = a b c , Ax = b. If det (A) ̸= 0, then the first entry of x is equal to A. −b + 17c − 6a B. 5/(−b + 17c − 6a) C. 0 D. 1 E. 2b − 5a 15. Let[2 3 1 −1 Find x1. ] [ x1 x2 ] = [3 −1 ] . by Cramer’s rule. x1 = 3 3 −1 −1 |A|= 0, x2 = 1 16. Find c1, c2 so that c1 [1 −2 ] + c2 [3 −4 ] = [−5 6 ] 17. Which of the following V is a vector space? A. V : the set of vectors B. V : the set of vectors [x y [x y ] with x ≤ 0, with the usual operations in R2. ] with x + y = 1, with the usual operations in R2. 3C. V : the set of all matrices [a b b + 1 d ] with regular matrix operaitons. D. V : the set of all positive real numbers, with ⊕ defined by u ⊕ v = uv, and c ⊙ u = uc. E. V : all n × n nonsingular matrices. 18. Is A = [5 1 −1 9 ] in span {[ 1 −1 0 3 ] , [1 1 0 2 ] , [2 2 −1 1 ]}? Are these four matri ces linearly dependent? 19. For what values of c are the vectors [−1, 0, −1] , [2, 1, 2] , [1, 1, c] linearly independent? 20. Find a basis for V = span 1 2 2 , 3 2 1 , 11 10 7 , 7 6 4 . Find a basis for V and a basis for V⊥. For V , use the reduced row echelon form: 1 3 11 7 2 2 10 6 2 1 7 4 → a 1 0 2 1 0 1 3 2 0 0 0 0 b 21. Find a basis for V : all vectors of the form c where b = a + c. 22. Find a basis for all 2 × 2 symmetric matrices. 23. Find a basis for span {t3 + t2 + 2t + 1, t3 − 3t + 1, t2 + t + 2, t + 1, t3 + 1} 24. Find a basis for the solution space of [1 1 1 1 2 1 −1 1 25. Decide the row/col rank and nullity of ] x = 0 1 −1 2 3 2 6 −8 1 2 −1 0 1 4 26. For what r does[1 r 2 1 have a nontrivial solution? ] x = 0 27. For a m×n matrix, discuss the max/min possible rank, nullity, and number of linearly independent rows/cols. 28. Find the angle between [−1/√2, 0, a], [0, −1, b]. For what a, b are they orthogonal? Orthonormal? 29. For what a, b is [a, b, 2] orthogonal to both [2, 1, 1] and [1, 0, 1]? 30. Orthonormalize [2, 1, 1] and [1, 0, 1]. 31. Find an orthonormal basis for the subspace of all vectors of the form [a, a + b, b]T. 2 1 , 32. Let V be a subspace of R3spanned by 1 2 . Find a basis for V⊥. For 4 −2 2 b = , find projV b and the distance between b and V . 3 1 33. Find a least squares soln to Ax = b 1 −1 −1 , b = 7014 A = 1 0 0 1 −1 0 0 1 −1 7 34. Find the least squares fitting for the points (1, 2),(2, 5),(3, 4),(4, 4) using A. Straight line. B. Quadratic polynomial. 35. Which of the following functions are linear transformations: A. L : R2 → R3 defined by L ([u1, u2]) = [u1 + 1, u2, u1 + u2]. B. L : R3 → R3 defined by L ([u1, u2, u3]) = [1, u2, u1]. C. L : R3 → R3 defined by L ([u1, u2, u3]) = [0, u3, u2]. D. L : Mnn → Mnn defined by L(A) = AT. E. L : Mnn → R defined by L(A) = det(A). 5F. L : Mnn → R defined by L(A) = A−1. G. L : M22 → R defined by L(A) = a11 + a22. ( 36. Find the standard matrix representation for L : R2 → R2 defined by L [u1 − 3u2, 2u1 − u2, 2u2]T. 37. A linear transformation L : M22 → M22 satisfies [u1, u2]T)= L L ([ 1 1 0 0 ([ 1 1 0 1 ]) = ]) = [0 1 2 1 [3 4 2 4 ] , L ] , L ([ 1 0 1 0 ([ 3 1 1 2 ]) = ]) = [5 1 2 3 [1 1 1 1 ] , ] . Find L ([ 4 5 6 7 ]). 38. Consider the matrices [1 1 1 1 ] , [2 1 1 2 ] , [1 −1 2 4 ] , 2 2 3 1 2 1 , 2 −2 1 2 1 0 1 2 1 0 1 2 , 0 −4 0 1 0 0 0 1 0 , 3 0 0 2 3 0 0 0 3 . A. Find the eigenvalues, eigenvectors, basis for the eigenspaces. B. Decide if they are diagonalizable or not. C. If yes, find P such that P−1AP = D is diagonal. D. Can P be orthogonal? 39. Let A = 2 0 0 0 cos α sin α 0 − sin α cos α . Find A−1. 40. Let B = P−1AP. Show that if x is an eigenvector of A then P−1x is an eigenvector of B. 41. Consider A = [2 + 2i −1 + 3i −2 1 − i ] , (or [3 −1 + 3i −1 − 3i 1 ] ) A. Find A−1. 6B. Find |A|. C. Solve Ax = b where b = [2 + i 2 − i ] . D. Find the eigenvalues and eigenvectors of A. 42. Let A = −2 −2 3 0 −2 2 0 2 1 A. Is A diagonalizable? If yes, diagonalize it. B. Solve x′ = Ax. Eigenvectors/eigenvalues: 3 1 0 0 ↔ −2, 1 21 21 ↔ 2, −7 −2 1 ↔ −3. C. For x′ = Ax, if x(0) = 2 1 , find x(1). 7