Midterm 1 - Review - ProblemsPeyam Ryan TabrizianTuesday, September 12th, 20111 Linear equationsProblem 1:Solve the following system (or say it has no solutions):x + 2y − z = 2x + 2y − 2z = 02x + 4y − 2z = 1Problem 2Use the following LU factorization of A to solve the equation Ax = b:A =1 32 7=1 02 11 30 1, b =102 Matrix products and inversesProblem 3Calculate AB (or say it’s undefined), where:(a) A =10, B =1 00 2(b) A =1 23 4, B =1 0 00 0 1Problem 4Find the inverse of the following matrix:A =1 0 30 1 01 1 11Problem 5Does the inverse of the following matrix exist?A =1 0 10 1 10 1 1Hint: This is a one-liner!3 Linear TransformationsProblem 6Assume T : R3→ R3maps e1to210, e2to3−10and e3to540. Find thematrix of T .Problem 7Assume T : R2→ R2rotates points in the plane by3π2radians. Find the matrixof T .4 N ul(A), Col(A), Linear dependence, SpanProblem 8(a) For the following matrix A, find a basis for Nul(A), Col(A).(b) Are the columns of A linearly independent? Do they span R4?A =3 −1 7 3 9−2 2 −2 7 6−5 9 3 3 4−2 6 6 3 7f3 −1 7 0 60 2 4 0 30 0 0 1 10 0 0 0 025 True/False ExtravaganzaProblem 9(a) If A is a 3 × 2 matrix, then Ax = 0 always has a nontrivial solution.(b) If T : Rn→ Rnis a linear transformation that is also onto, then theequation Ax = b has a solution for every b.(c) If AB = I, then A is invertible(d) If A and B are 2 × 2 matrices such that A 6= O and B 6= 0, then AB 6= O(where O is the zero matrix)(e) If A is n × n and has n pivot rows, then the columns of A span Rn(f) If A is invertible, then Nul(A) = {0}(g) If v1, v2, v3are in R4and v3= 2v1+ v2, then {v1, v2, v3} is linearlydependent.(h) If v1, v2, v3, v4are in R4and {v1, v2, v3} is linearly dependent, then{v1, v2, v3, v4} is linearly
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