Math 21 Winter 2017 Professor Bruce Cooperstein Final Review/Practice Examination You will need to know the definitions of the following in order to do the questions on the examination. You will not be asked to explicitly state them but nearly every questions will implicitly test whether you understand the meaning of some fundamental concept. i. A linear combination of a sequence of vectors v1 , v2 , . . . , vk . ii. The span of a sequence of vectors v1 , v2 , . . . , vk . iii. The vectors v1 , v2 , . . . , vk span the vector space V. iv. The vectors v1 , v2 , . . . , vk are linearly independent. v. The vectors v1 , v2 , . . . , vk are linearly dependent. vi. The null space of the matrix A. vii. The column space of the matrix A. viii. The rank of a matrix A. ix. The nullity of a matrix A. x. Basis of a vector space V. xi. Dimension of a vector space V. xii. Subspace of a vector space V. xiii. A matrix A is invertible. The inverse of a matrix A. xiv. For B a basis of a vector space V and v a vector in V, the coordinate vector of v with respect to B. xv. For bases B and B 0 of a vector space V the change of basis matrix PBB0 . xvi. For vector spaces V and W a linear transformation T from V to W. xvii. For V and W vector spaces, an isomorphism T from V to W. xix. For T : V W a linear tranformation, the kernel of T, Ker(T ). xx. For T : V W the range of T, R(T ). xxi. For T : V W a linear tranfsormation, the nulllity of T, N ullity(T ). xxii. For T : V W a linear tranformation, the rank of T, Rank(T ). xxiii. Let V be a vector space with a basis BV = (v1 , . . . , vn ) and W a vector space with a basis BW = (w1 , . . . , wm ) and T : V W a linear transformation, the matrix of T with respect to (BV , BW ), MT (BV , BW ). xxiii. For a square matrix A an eigenvector v for A with eigenvalue . 1 1. Each of the following sequences is linearly dependent in R3 . Give a succinct reason why this is so. a) The columns of a 3 4 matrix. 2 b) The columns of a 3 3 matrix A where null(A) = span(1). 3 c) If A is a 3 3 matrix and x is a 3-vector, the sequence (x, Ax, A2 x, A3 x). 2. Without computing explain why the following are linearly dependent sequences a) The sequence (1 + 2x, 1 x2 , 1 + 2x, 1 + x3 ) in R3 [x]. b) For a 3 3 matrix A, the sequence (I3 , A, A2 , A3 , A4 , A5 , A6 , A7 , A8 , A9 ) in M33 (R). 2 1 1 1 1 2 1 2 3. Determine if the vector 3 is in Span( 1 , 1 , 1 ). 4 4 3 6 4. a) Let 1 0 A= 3 1 1 0 3 1 1 1 2 1 1 3 . 4 4 1 0 0 2 Without computing, give a succinct reason why the columns of A do not span R4 . b) Assume that the matrix B is row equivalent to the matrix 1 1 0 2 0 4 0 0 1 3 0 5 0 0 0 0 1 6 Explain why the first, third, and fifth columns of B span R3 . 5. Assume f1 , f2 , f3 , f4 are in R2 [x]. a) Explain why (f1 , f2 ) is not a basis of R2 [x]. b) Explain why (f1 , f2 , f3 , f4 ) is not a basis of R2 [x]. 1 2 1 6. Let v1 = 2 , v2 = 5 , v3 = 1 and set B = (v1 , v2 , v3 ). Determine 0 1 2 if B is a basis for R3 . Whatever you determine you should begin your conclusion by \"B is (is not) a basis because ...\" 2 1 1 7. a) Determine if the vector 1 is 1 1 2 2 1 3 2 1 2 in the null space of the matrix 3 1 2 1 6 0 . 1 1 b) The 1 1 2 3 2 1 reduced row echelon form of the augmented matrix 2 1 | 2 1 0 3 2 | 3 3 1 | 3 is 0 1 1 1 | 1 . 5 3 | 5 0 0 0 0 | 0 1 1 2 1 Find a basis for the null space of 2 3 3 1. 2 1 5 3 c) Express the general solution to 1 1 2 3 2 1 the matrix equation 2 2 1 3 1 x = 3 5 5 3 in the form p + S where p is a particular solution and S is a subspace of R4 . 8. Find the inverse of the matrix 1 2 3 2 5 7 1 3 5 9. Find the determinant of each of the following matrices. In some instances by making use of the properties of matrices you can find the determinant without the necessity of extensive computation. When doing so, state what properties you are using. 1 2 3 4 5 5 4 3 2 1 a) 0 0 0 0 0 . 1 2 1 2 1 1 2 3 4 1 3 4 5 b) 2 6 6 6 1 1 3 7 1 2 1 3 3 8 1 1 c) 1 6 0 4 2 0 1 3 3 10. A, B, C are 4 4 matrices. Assume the following; i. det(A) = 4 0 1 2 0 ii. B 3 = 0 . 0 4 iii. det(C) = 3 Using properties of determinants compute the following: a) det(3A) c) det(A2 B) d) det(AC 2 ) e) det(C 1 AT ) 11. V is a 4-dimensional vector space with bases B = (v1 , v2 , v3 , v4 ) and B 0 = (w1 , w2 , w3 , w4 ) where v1 = w1 w2 + w4 v2 = w1 + w3 w4 v3 = w1 + w2 + w3 2w4 v4 = 2w1 + w2 + 2w3 2w4 . a) Write down [v1 ]B0 , [v2 ]B0 , [v3 ]B0 , [v4 ]B0 . b) Write down PBB0 . 1 1 c) If [u]B = 1 what is [u]B0 ? 2 12. Let B = (f1 , f2 , f3 ), a basis of R2 [x], where f1 = 1 x f2 = x x2 f3 = 1 + x x 2 . a) If g = 3f1 + 2f2 5f3 determine [g]B . 2 b) If [h]B = 1 determine h (express as a polynomial in R2 [x]). 1 c) If S = (1, x, x2 ) determine the change of basis matrix PBS . 4 d) Determine the change of basis matrix PSB . e) Determine [1 + 2x + 3x3 ]B . 1 1 1 1 1 2 0 3 13. Let A = 1 1 1 1. 3 4 2 4 a) Find a subsequence B of the columns of A which is a basis for col(A). b) Extend B to a basis of R4 . 14. Let T : M22 (R) R4 be a linear transformation. \u0012 \u0013 \u0012 1 0 0 a) Assume that Ker(T ). Prove that T ( 0 2 0 \u0013 \u0012 0 1 1 ) = 2T( 1 0 \u0013 0 . 0 b) Continue with the assumption of a). Explain why T is not surjective. 15. Assume B = (v1 , v2 , v3 , v4 ) is a basis for a vector space V and T : V V is a linear transformation which satisfies T (v1 ) = v1 v2 + v3 v4 T (v2 ) = 2v1 3v2 + 3v3 v4 T (v3 ) = 2v1 + 3v2 2v3 + v4 T (v4 ) = v1 2v2 + 4v3 3v4 . a) What is [T (v2 )]B ? b) What is the matrix of T with respect to B, MT (B, B)? 16. Let V be a vector space with basis BV = (v1 , v2 , v3 , v4 ) and W a vector space with basis BW = (w1 , w2 , w3 ). Let T be the linear transformation from V to W which satisfies T (v1 ) = w1 + 2w2 + 3w3 T (v2 ) = w1 + 3w2 + 3w3 T (v3 ) = w1 + w2 + 4w3 T (v4 ) = w2 w3 a) Write down the matrix of the linear transformation T with respect to the bases BV and BW . 1 1 b) If [u]BV = 2 find [T (v)]BW . 2 5 1 2 c) If [v]BV = 1 determine if v is in the kernel of T. 3 1 determine if w is in the range of T. 1 d) If [w]BW = 4 17. Let T : M22 (R) R2 [x] be the linear transformation such that the matrix of T with respect to the standard basis of M22 (R) and the standard basis of 1 2 1 1 R2 [x] is 2 3 1 3. 1 1 0 2 a) Find a basis for the kernel of T, Ker(T ) (these should be vectors in M22 (R). b)Find a basis for the range of T , R(T ) (these should be vectors in R2 [x]. 2 1 1 2 18. a) The vector x = 1 is an eigenvector of the matrix 1 1 1 1 1 1 . 2 Determine the corresponding eigenvalue. 2 b) The matrix 1 3 2 2 1 2 has 3 as an eigenvalue. 6 3 Determine a basis for the corresponding eigenspace E3 . \u0012 19. Let A = 5 4 \u0013 2 . 4 a) Find the characteristic polynomial of A. b) Determine the eigenvalues of A. c) For each eigenvalue find a basis for the eigenspace corresponding to the eigenvalue. 6