61 Name Math 270 S19 Key Pearsall EXAM 1) Answer TRUE or FALSE. Explain your answer: True a) A matrix A is not invertible if and only if 0 is an eigenvalue of A. Statement from see 5 True vector b) Finding an eigenvalue of A may be difficult, but checking whether a given vector is in fact an eigenvector is easy. statement from see 5 False of the entries c) A determinant of A is the product diagonal in A. True for a diag. mathix False d) For a square matrix A, vectors in Col A are orthogonal to vectors in Nul A. If A this is not Arue. False e) A is diagonalizable if and only if A has n eigenvalues, counting multiplicities. Example from see. 5. False f) A matrix with orthonormal columns is an orthogonal matrix. Has to be a square matine. 12 61 Name Math 270 S19 Key Pearsall EXAM 1) Answer TRUE or FALSE. Explain your answer: True a) A matrix A is not invertible if and only if 0 is an eigenvalue of A. Statement from see 5 True vector b) Finding an eigenvalue of A may be difficult, but checking whether a given vector is in fact an eigenvector is easy. statement from see 5 False of the entries c) A determinant of A is the product diagonal in A. True for a diag. mathix False d) For a square matrix A, vectors in Col A are orthogonal to vectors in Nul A. If A this is not Arue. False e) A is diagonalizable if and only if A has n eigenvalues, counting multiplicities. Example from see. 5. False f) A matrix with orthonormal columns is an orthogonal matrix. Has to be a square matine. 12 True g) If is in a subspace W, then the orthogonal projection of y y onto W is y itself. Statement from Sec. 6. True If there has orthonormal then h) A QR, Q columns, Problem done n class from see 6. True An that is must i) n X n matrix orthogonally diagonalizable be symmetric. Then in Sec 7 True A matrix of form is matrix. j) a quadratic a symmetric Ref. in Sec. 7. 2pts. 332 73 2) Is an eigenvalue of Why or why not? 3 3 in 9 (x 2, X2 Pyes 10 True g) If is in a subspace W, then the orthogonal projection of y y onto W is y itself. Statement from Sec. 6. True If there has orthonormal then h) A QR, Q columns, Problem done n class from see 6. True An that is must i) n X n matrix orthogonally diagonalizable be symmetric. Then in Sec 7 True A matrix of form is matrix. j) a quadratic a symmetric Ref. in Sec. 7. 2pts. 332 73 2) Is an eigenvalue of Why or why not? 3 3 in 9 (x 2, X2 Pyes 10 3 4pt 3 5) Determine which set of vectors are orthogonal: u 1 3 4 3 8 W 7 0 u. v HIH 3 3 4 3 is . w 3 8 0 4 0 3 i. 13 3 7 3 8 : they are all orthogonal to each other 4 3 4pt 3 5) Determine which set of vectors are orthogonal: u 1 3 4 3 8 W 7 0 u. v HIH 3 3 4 3 is . w 3 8 0 4 0 3 i. 13 3 7 3 8 : they are all orthogonal to each other 4 4 assume is 1, is an orthogonal basis for IR 0 3 1 5 8pt orthogonal 1 5 0 6) Given u W and Z is a basis for , , 1 1 R 4 1 1 10 , and let X Write X as the sum of two vectors, one in 2 0 and the other in X 15 x W X v X w 2 16 u V 12 72 W 18 36 18 36 3 S 5 2 O 9 3 2 10 8 0 32 9 2 0 10 6 4 2 2 2 8 9 15 2 2 N 8 4 assume is 1, is an orthogonal basis for IR 0 3 1 5 8pt orthogonal 1 5 0 6) Given u W and Z is a basis for , , 1 1 R 4 1 1 10 , and let X Write X as the sum of two vectors, one in 2 0 and the other in X 15 x W X v X w 2 16 u V 12 72 W 18 36 18 36 3 S 5 2 O 9 3 2 10 8 0 32 9 2 0 10 6 4 2 2 2 8 9 15 2 2 N 8 2pto. 9) Compute the quadratic form XT AX, when This and 6 I 1 3 5 3 m 91 THE 3 V 185 3 10pts. 10) PROVE: If square matrices A and B are similar, then they have the same characteristic polynomial and hence the same eigenvalues (with the same multiplicities). Proof: If then det So A and B have the same eigenvalies. So det (det det P (det det(I) 12 2pto. 9) Compute the quadratic form XT AX, when This and 6 I 1 3 5 3 m 91 THE 3 V 185 3 10pts. 10) PROVE: If square matrices A and B are similar, then they have the same characteristic polynomial and hence the same eigenvalues (with the same multiplicities). Proof: If then det So A and B have the same eigenvalies. So det (det det P (det det(I) 12
