Review Linear Algebra October 22, 2013
Question 1 Let A and B be matrices. If AB is a 4 7 matrix, then determine the dimensions of A and B if A has 19 columns.
Answer 1 Answer A is a 4 19 matrix, while B is a 19 7 matrix.
Question 2 Determine whether the following statements are true or false. If the statement is true, explain why the given statement is true (a proof is not necessary), otherwise provide a counterexample. For every m n matrix A and 0 R n, the matrix equation A x = 0 has a nontrivial solution.
Answer 2 Answer False, the matrix equation 0 = I 3 x has only the trivial solution.
Question 3 State the definition of a linear transformation.
Answer 3 Answer A linear transformation is a transformation T : R m R n that satisfies the following properties 1 T ( u + v ) = T ( u ) + T ( v ) 2 T (c u ) = ct ( u )
Question 4 For the following problems determine if the given transformation is a linear transformation. Justify each answer by showing your work. Let T : R 3 R 4 be given by 2x x 1 + x 3 1 T x 2 = 3x 2 x 3 x x 1 x 2. 3 x 1 + x 2 + x 3
Answer 4 Answer T is a linear transformation (see board for explanation).
Question 5 State the definition of an onto transformation.
Answer 5 Answer A transformation T : R m R n is called onto if for every b R n there exists u R m such that T ( u ) = b.
Question 6 Let T : R 3 R 4 be given by 2x x 1 + x 3 1 T x 2 = 3x 2 x 3 x x 1 x 2 3. Determine if the x 1 + x 2 + x 3 aforementioned transformation is one-to-one and/or onto. Justify your answer.
Answer 6 Answer The given transformation is one-to-one but not onto (see board for explanation).
Question 7 Find integers for m and n in order for the following statements to be true. Let A be an m n matrix with x R 7 and b R 4. Then we can form the matrix equation A x = b.
Answer 7 Answer m = 4 n = 7
Question 8 Find integers for m and n in order for the following statements to be true. Suppose S : R 6 R 3 and T : R 3 R 19 are linear transformations. Then the linear transformation T S can be represented by an m n matrix.
Answer 8 Answer m = 19 n = 6
Question 9 State the definition of an invertible matrix.
Answer 9 Answer An n n matrix A is called invertible if there exists an n n matrix B such that AB = I n.
Question 10 Determine whether the following statements are true or false. If the statement is true, give justification as to why the given statement is true, otherwise provide a counterexample. If A and B be are invertible n n matrices, then A 1 BA = B.
Answer 10 Answer False, see board for counterexample.
Question 11 State the definition of a basis for a subspace.
Answer 11 Answer A basis for a subspace is a set of vectors that are linearly independent and span the given subspace.
Question 12 Determine whether the following statements are true or false. If the statement is true, give justification as to why the given statement is true, otherwise provide a counterexample. Let { v 1, v 2,, v k } be a basis for a subspace S. Then the set of vectors { v 1, v 2,, v k, w } is linearly dependent for every w S.
Answer 12 Answer This statement is true (see board for explanation).
Question 13 State the definition of an eigenvalue of a matrix A.
Answer 13 Answer An eigenvalue of a matrix A is a scalar λ such that there exists a nonzero vector v such that A v = λ v.
Question 14 The eigenvectors for a matrix A are given by { y, z } where [ ] 3 y = and [ ] 5 z = 9 7 with corresponding eigenvalues λ 1 = 3 and λ 2 = 2, respectively. Use this information to determine A 4 v where [ ] 3 v =. 15
Answer 14 Answer [ 732 2580 ].
Question 15 State the definition of a subspace of R n.
Answer 15 Answer A subspace S of R n is a set that satisfies the following properties: 1 0 S 2 If u, v S, then u + v S 3 If u S and c R, then c u S
Question 16 Determine if the collection of vectors of the form x 2 + y 2 x + y x is a subspace of R 3.
Answer 16 Answer This is not a subspace (see the board for the explanation).