Math 191 Sample Midterm 2A Spring 2023
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1a. (5 points) Suppose A is an n × n matrix with complex entries and A∗ = A. Suppose V is a subspaceofCn suchthatAV ⊂V,i.e.,x∈V ⇒ Ax∈V. ShowthatA(V⊥)⊂V⊥.
1b. (5 points) Suppose A is a real, symmetric, n × n matrix. Show that eA is positive definite.
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2a. (2 points) Write down the overdetermined linear system Ax = b whose least squares solution C
x = D gives the best-fit line y(t) = C + Dt to the following points (ti, bi) ti −4/3 1 4 5
bi − 13/4 −1/4 9/4 13/4 in the sense that ∥r∥2 is minimized, where ri = bi − y(ti).
2b. (5 points) Find the Householder transformation H1 = I − τ1v1v1T that reflects the first column of the matrix from part (a) to lie along the b1 axis. (Find τ1 and v1, following the convention that (v1)1 = 1.)
2c. (3 points) After applying H1 and computing and applying a second householder transformation, the above system becomes
00 −0.4 Compute C, D and the norm ∥r∥2 of the minimum residual.
−2 −13/3 −1 0 −5C−5
0 0 D =−0.3.
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3. (10 points) Let A = 4 −3 . Find all rank-1 matrices B such that ∥A − B∥2 is minimized.
Hint: if you can’t figure out the SVD by inspection, AAT is simpler than AT A as a starting point to compute the SVD systematically.
4. (10 points) Compute the pseudo-inverse of
1 21 0 1 1 2 A=−2 2 2 1 −2 0 1 .
You can leave your answer as a product of 3 matrices if you wish, but compute each entry of each of those matrices.
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