Math 2B Spring 3 Final Exam Name Write all responses on separate paper. Show your work for credit.. True or false, with reason if true and counterexample if false: a. Every invertible matrix can be factored as where is lower triangular and has all s on its diagonal and is upper triangular with the pivots on its diagonal. Before 2 0 confirming your answer, consider 2 4. What s going on? b. A matrix can be factored as, where is orthonormal and is upper triangular, only if is non-singular. c. Every invertible matrix can be diagonalized. d. Every diagonal matrix can be inverted. e. Exchanging the rows of a 22 matrix reverses the signs of its eigenvalues. 4 2. Consider A 0. a. Factor as a product elementary matrices where is a permutation matrix, is lower triangular with s on the diagonal, is upper triangular with s on the diagonal and is upper triangular with s on the diagonal (not that is such a matrix). Use these to interpret the transformation geometrically as some combination of dilations/contractions, shears and/or reflections. b. Factor A as a product of an orthogonal matrix Q and an upper triangular matrix R. Hint: First use gram-schmidt to find and then compute.. a. Project b 2 2 2 2 onto each of the orthonormal vectors u,, and u,, and 3 3 3 3 3 3 find its projection onto the plane of u and u 2. 3. Consider b 0,3,0 b. Project b 2 2 onto u3,, and add this projection to the two projections you found 3 3 3 in part (a). How do you interpret the result? 4. If is a unit vector, show that 2 is an orthogonal matrix. Compute when,,,.
0 5. Let 0 2 a. Draw a diagram illustrating the column space of A. Hint draw a vector in the xz-plane and a vector in the yz-plane and then a third vector from the endpoint of one to the endpoint of the other this is a triangle contained in the plane of the column space. b. Draw a diagram illustrating the row space of A. Hint: Draw a basis vector in the xy-plance and a basis vector in the yz-plane and then a vector from the endpoint of one to the endpoint of the other this is a triangle contained in the plane of the row space. c. Draw a diagram illustrating the null space of A. d. Draw a diagram illustrating the left null space of A. 2 e. Let 2. Find the complete solution to. 4 6. Find the eigenvalues and eigenvectors and a diagonalizing matrix for A 7 2 5 4. 7. Find the determinants of A and A - if A P 0 0 P. 2 8. Suppose that A has eigenvalues 0 and, corresponding to the eigenvectors and 2. a. How can you tell in advance (without actually computing A) that A is symmetric? b. What are the trace and determinant of A? c. What are the eigenvalues, eigenvectors and determinant of A 2? 9. Find k 0.4 0.3 a lim k 0.6 0.7 b 0. Consider the network shown at right. Whose incidence matrix is 0 0 0 0 0 0 0 0 0 0 a. Find the incidence matrix for this network. b. Compute and find its eigenvalues. c. Compute the eigenvectors for.
Math 2B Spring 3 Final Exam Solutions. True or false, with reason if true and counterexample if false: a. Every invertible matrix can be factored as where is lower triangular and has all s on its diagonal and is upper triangular with the pivots on its diagonal. Before 2 0 confirming your answer, consider 2 4. What s going on? SOLN: This is almost true, with the exception that sometimes a permutation matrix is required. The proposed example is factored like so: 2 0 0 0 0 0 2 0 2 4 0 0 00 0 0 2 0 0 0 b. A matrix can be factored as, where is orthonormal and is upper triangular, only if is non-singular. SOLN: No. Here s a quick counterexample: 0 0 0 0 0 0. c. Every invertible matrix can be diagonalized. SOLN: No. It could be the matrix has a repeated eigenvalue with only one eigenvector. For example if 3 0 3 then the characteristic equation is 3 0 so there is only one eigenvector,0, so you can t diagonalize. d. Every diagonal matrix can be inverted. SOLN: No. If there s a zero on the diagonal then the matrix is singular. e. Exchanging the rows of a 22 matrix reverses the signs of its eigenvalues. SOLN: No. Suppose the rows are the same. Then swapping rows won t change anything. 2. Consider 4 0. a. Factor as a product elementary matrices where is a permutation matrix, is lower triangular with s on the diagonal, is upper triangular with s on the diagonal and is upper triangular with s on the diagonal (not that is such a matrix). Use these to interpret the transformation geometrically as some combination of dilations/contractions, shears and/or reflections. SOLN: 4 0 0 4 0 4 0 0 0 4 4 0. This first shears by a factor of 4 parallel to the x-axis. Then stretch by a factor of 4 in the y direction while reflecting across the x-axis. Finally, a shear by a factor of parallel to the x-axis. b. Factor A as a product of an orthogonal matrix Q and an upper triangular matrix R. Hint: First use gram-schmidt to find and then compute. SOLN: Using the gram-shmidt formula (not really needed here, but it shows good practice) 4 0 2 2. So 2/2 2/2 and 2 2/2 2/2 4 2 2 4 2/2 2 2 2. So 2/2 2 0 0 2 2 0 2/2 2/2 0 2 2
3. Consider 0,3,0 a. Project onto each of the orthonormal vectors,, and 2,,. and find its projection onto the plane of u and u 2. SOLN: The projection of onto is 2,, and the projection of onto is 2,,. The projection of onto the plane of and is the sum of these vectors: 2,, 2,,,,. b. Project onto,, and add this projection to the two projections you found in part (a). How do you interpret the result? SOLN:,,,,,,,,. 4. If is a unit vector, show that is an orthogonal matrix. Compute when,,,. SOLN: My apologies There was a typo in the problem statement which I didn t notice and wasn t called during the exam. That s unfortunate. The orthogonal matrix is 2. 2 2 2 2 224 shows the intended matrix is orthonormal, however, the given matrix has 0 indicating that it s a projection onto the null space of something maybe, but it s not an orthonormal matrix! When,,,, 2 That s orthonormal.. Note this is not orthogonal. Look at
0 5. Let 0 2 a. Draw a diagram illustrating the column space of A. Hint draw a vector in the xz-plane and a vector in the yzplane and then a third vector from the endpoint of one to the endpoint of the other this is a triangle contained in the plane of the column space. SOLN-> b. Draw a diagram illustrating the row space of A. Hint: Draw a basis vector in the xy-plance and a basis vector in the yz-plane and then a vector from the endpoint of one to the endpoint of the other this is a triangle contained in the plane of the row space. SOLN: c. Draw a diagram illustrating the null space of A. 0 0 SOLN: 0 ~0 and so the null space is the span of the vector,,. 2 0 0 0
Note that the is a line orthogonal to the plane of the row space,. d. Draw a diagram illustrating the left null space of A. 0 0 SOLN: 2~0 and so the left null space is the span of,,. 0 0 0 0 Note that the left null space is orthogonal to the column space. 2 e. Let 2. Find the complete solution to. 4 0 2 SOLN: 0 2 2 4
6. Find the eigenvalues and eigenvectors and a diagonalizing matrix for 7 2 5 4. SOLN: 74 30 322 the eigenvalues are 2 and. For 2 the eigenvector is 2, 5 and for the eigenvector is,3 Thus 2 5 3. To be sure, Λ 2 5 3 2 0 0 3 5 2 2 5 3 6 2 5 2 7 2 Nifty. 5 4 7. Find the determinants of A and A - if A P 0 0 P. SOLN: and 2 8. Suppose that A has eigenvalues 0 and, corresponding to the eigenvectors and 2. a. How can you tell in advance (without actually computing A) that A is symmetric? SOLN: The diagonalizing matrix is (anti) symmetric. That is, 2 2 0 0 0 2 2 2 2 0 0 4 2 2 2 is a product of (anti) symmetric matrices, and so must itself be symmetric. Also, note that Further, note that it is *not* sufficient for A to have different eigenvalues/vectors. For 0 example, A is not symmetric and yet has different eigenvectors and eigenvalues. 0 b. What are the trace and determinant of A? SOLN: Since A has a zero eigenvalue, the determinant is zero. The trace is the sum of eigenvalues =. a. What are the eigenvectors and eigenvalues of A 2? SOLN: 2 2 0 0 0 2 2 2 2 0 0 4 2 2 2, so it has the same eigenvalues and eigenvectors. 0.4 0.3 9. Find lim 0.6 0.7 SOLN: The characteristic equation: 0.4 0.3 0.6 0.7 2 5 7 0 9 50 0 0 0 shows eigenvalues are and /0 with corresponding eigenvectors 2 and. Thus 0.4 0.3, lim 0.6 0.7 lim 2 0 0 0. 2 2 0 0 0 2 2 0 0 2 2 2 2
0. Consider the network shown at right. Whose incidence matrix is 0 0 0 0 0 0 0 0 0 0 a. Compute and find its eigenvalues. 0 0 0 0 0 0 3 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 3 0 2 3 2 0 3 0 2 2 3 3 3 2 2 2 0 0 3 232232 23232223 2822 3 20 3 2 3 28 5 2296 85 2 8 6 4 2 So the eigenvalues are 4 (twice), 2 and 0. b. Compute the eigenvectors for. SOLN: 0 2 For 4 we have 2 0 0 0 4 ~ So it has two 0 0 0 0 2 0 2 0 0 0 0 0 eigenvectors:, 0 0
0 0 0 For 2 we have 0 0 0 0 2 ~ so this is the 0 0 0 0 0 0 0 0 0 0 eigenvector and, finally, for 0 we have 0 3 0 0 2 0 0 0 0 ~ and eigenvector 3 0 0 0 2 0 0 0 0 As a bonus, we have the diagonalization 3 2 0 3 2 0 4 0 0 0 3 4 0 0 4 0 0 0 0 0 0 2 0 0 2 0 2 0 2 0 0 0 0 0