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Math 3130 - Intro to Linear Algebra - MathOnline - Spring 2016 Instructor: Dr. Radu C. Cascaval Department of Mathematics - University of Colorado at Colorado Springs Email: [email protected], Fax: 1-888-951-4321 (toll free) To be read and completed by the Proctor: By signing below, I certify that the exam was administered in the time allowed (75 minutes) and observing the following rules: no books/notes; scientific calculator (WITHOUT graphic or matrix capabilities) is allowed. Proctor Name: Proctor Signature (required) (Please fax or email the completed exam no later than Wed, April 20, 5pm MT.) Student Name: Exam 2 - Math 3130 - Intro to Linear Algebra The exam has 8 problems on 5 pages. Make sure you turn in ALL pages. To receive credit, show all work. No calculators with linear algebra capabilities allowed! Exercise 1.[3 pts each] Determine whether each of the following statements are TRUE or FALSE. Circle your answer and justify it for credit. No justification, no credit. • The set of nonzero row vectors of a matrix A is a basis for the row space of A TRUE FALSE • The reduced row echelon form of AT is the transpose of the reduced row echelon form of A. TRUE FALSE • Every linearly independent subset {v1 , . . . , vn } of a n-dimensional vector space is a basis. TRUE FALSE • The span of any finite set of vectors in a vector space V is a subspace of V . TRUE FALSE Exercise 2.[5 pts each] Decide whether Cramer’s rule can be used in solving each of the linear system below or not. If yes, solve it using Cramer’s rule. If not, explain why. ( 3x1 2x2 = 1 (a) x1 2x2 = 2 8 > < x + 2y + 3z (b) x y + z > : x + 5y + 5z =1 =2 =0 Exercise 3.[5 pts each] (a) Find the complete solution of the linear system 2 1 2 A = 42 6 0 2 Ax = b, where 3 2 3 1 2 35 , and b = 445 1 0 (b) What is the rank of the matrix A? What is the column space of A? Exercise 4.[5 pts each] (a) Find a basis for the subspace of R3 spanned by the following 4 vectors 2 3 2 3 2 3 2 3 1 0 2 1 v1 = 4 35 , v2 = 415 , v3 = 4 7 5 , v4 = 405 0 2 2 1 (b) Express all the remaining vectors as linear combinations of the basis found in (a). Exercise 5.[9 pts] Find a basis for the subspace of R4 consisting of all vectors that are orthogonal to v1 = (1, 1, 2, 1)T and v2 = (0, 0, 1, 3)T . Exercise 6.[6 pts each] Given the matrix (1) Determine the rank of A. 2 1 A = 41 0 (2) Find a basis for the column space of A. (3) Find a basis for the row space of A. (4) Find a basis for the nullspace space of A. 2 2 0 1 0 1 4 5 1 3 1 05 1 Exercise 7.[5 pts each] Find a basis and dimension of the following subspaces of R4 (a) All vectors of the form (a, b, c, d) with a + b + c + d = 0 (b) All vectors of the form (a, b, c, d) with a = b = c = d. Exercise 8.[5 pts each] Determine which of the following are subspaces of Mn⇥n , the space of all n ⇥ n matrices and which are not. Find the dimension of these subspaces. (a) The set of all diagonal n ⇥ n matrices. (b) The set of all n ⇥ n matrices A such that det(A) = 0. (c) The set of all symmetric matrices A (AT = A)