Preparation problems for the discussion sections on Novemb er 11th and 13th1. Let A =1 00 11 1.a. Find the QR decomposition of A: write A = QR where Q is a matrix with orthonormalcolumns and R is an upper triangular matrix.b. Let b =001. Use the QR decomposition of A to find the least squares solution ofAbx = b (by solving Rbx = QTb).2. a. Compare det1 23 4and the “row flipped” determinant det3 41 2.b. If A =0 0 0 0 10 0 0 1 00 0 1 0 00 1 0 0 01 0 0 0 0, what is det(A)?c. If A =1 1 42 2 53 3 6, what is det(A)?d. If A =1 4 52 5 73 6 9, what is det(A)?e. If A, B are 3 × 3 matrices with det(A) = 2, det(B) = −1, calculate(i) det(BAT),(ii) det(BAB−1),(iii) det(A−1).f. If A =1 2 33 2 11 1 3, find det(A) by expanding along the last column.3. a. Someone tells you that det is linear, so det(3A) = 3 det(A). What do you answer?(What about det(31 00 1)? If A is a 3 × 3 matrix, and det(A) = 2 what is det(3A)?)b. Somebody tells you that the matrixA =1 2 −2 02 3 −4 0−1 −2 0 00 2 5 0is invertible. What do you say?c. LetA =1 2 −2 02 3 −4 1−1 −2 0 20 2 5 3.Calculate det(A). Is A invertible?1d. Let A be a 3 × 3 matrix so that A11−1= 0. What is det(A).4. Reading through your favorite linear algebra textbook, you find the following interestingstatement: if the columns of A are independent, then the orthogonal projection onto ColA hasprojection matrix A(ATA)−1AT.a. How does this formula simplify in the case when A has orthonormal columns?b. Let Q =1 00350 −45. What is the projection matrix corresponding to the orthogonalprojection onto Col(Q)?c. Let Q =1 0 0035450 −4535. What is the projection matrix corresponding to the orthogonalprojection onto Col(Q)? Explain why your answer is not surprising.d. (optional) Can you explain the formula A(ATA)−1ATfor the projection matrix usingthe normal equations for least squares?5. True or False? Justify your answers!a. Let Q be a 3 × 3 orthogonal matrix. Then det(Q) = 1.b. If det(A) =det(B) = 0 then det(A + B) = 0.c. We say A and B (n × n matrices) are similar if A = DBD−1for an invertible matrixD. Let A and B be similar matrices, then det(A) =det(B).d. Let A and B be 3 × 3 matrices. If det(A) =det(B) then A and B are similar. [Note:number of pivots in DBD−1is equal to the number of pivots in B. (Why?) Use thisfact to find a counter example.]e. Let A be a 3 × 3 matrix so that det(A) = 0. Then Ax = b has exactly one solution foreach vector b.f. Let A be a 3 × 3 matrix so that det(A) = 9. Then det(2A) = 18.g. Let R be a 2 × 3 matrix. Then det(RTR) = 0.h. Let R be a 2 × 3 matrix. Then det(RRT) = 0.6. Let f be a function with period 2π that satisfies f (x) = x on (−π, π]. Find the Fourierseries of f
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