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MATH 221 - EXAM #2 Name: Student ID: Exam rules: • No calculators, open books or notes are allowed. • There are 9 problems in this exam. The number of points for each problem is given in the brackets. 1 Problem 1. [5 Pts] Find the eigenvalues of the matrix below. For each eigenvalue indicate its algebraic multiplicity. 1 0 2 −1 0 1 −1 2 0 0 1 0 −1 2 1 5 Problem 2. [] The matrix A below has eigenvalues λ1 = −1, λ2 = 2. Find a basis for R3 consisting of eigenvectors of A. 0 1 1 A = 1 0 1 1 1 0 Problem 3. [] Find the determinant of the matrix: 1 −1 5 5 2 3 1 2 4 5 −1 −3 8 0 −1 1 1 2 −1 2 −1 1 −3 1 −1 Problem 4. [9] Find all values of b such that the matrix A is NOT invertible: 0 b −1 0 b 0 0 2 A= 0 1 2 1 5 0 0 5 Problem 5. [5] Find a basis for Span{~v1 , ~v2 , ~v3 , ~v4 , ~v5 }, where 1 1 −2 3 2 −1 3 4 ~v1 = −1 , ~v2 = −2 , ~v3 = 3 , ~v4 = 4 , 3 −1 2 14 2 1 ~v5 = 0 . 5 Problem 6. Let T : R3 → R3 be the linear transformation T (x1 , x2 , x3 ) = (x1 − 2x2 + 2x3 , 2x1 + x2 + x3 , x1 + x3 ). a.[3] Find the matrix of the linear transformation T . b.[3] Find the inverse transformation T −1 expressed the same way as T . x1 x2 Problem 7. [10] Let H be the subspace of R5 consisting of all vectors x3 that x4 x5 satify the four linear equations: x1 − 3x2 + 2x4 + 2x5 = 0 −2x1 + 6x2 + x3 + 2x4 − 5x5 = 0 3x1 − 9x2 − 1x3 + 7x5 = 0 −3x1 + 9x2 + 2x3 + 6x4 + −8x5 = 0 Find a basis for H. Problem 8. [10] a.[4] Let A and B be some fixed m × n matrices. Prove that H = {all ~v in Rn such that A~v = B~v } is a subspace of Rn . b.[2] Let H be the set R2 with the x-axis removed, but with ~0 put back to H: x H={ : y 6= 0} ∪ {~0}. y Show that H is not a subspace of R2 . (Find one instance where some axiom does not hold.) Problem 9. [10] Mark each statement either True or False. You do not have to justify your answer. a.For any m × n matrix A and vector ~b in Rm , the set of solutions of A~x = ~b is a subspace of Rn . b.If v is an eigenvector of A then v is also an eigenvector of A2 . c.For matrices A and B, if AT B is defined, then AB T is also defined. d.If A and B are 3 × 2 matrices then A · B T always has λ = 0 as an eigenvalue. e.Let A be a n × n matrix, {~v1 , . . . , ~vk } a basis for N ul(A) and {w ~ 1, . . . , w ~ l} a basis for Col(A). For some A it is possible that {~v1 , . . . , ~vk , w ~ 1, . . . , w ~ l } forms a basis for Rn .