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1. (a) Determine whether the following three vectors v, = (0, 3, 1,-1), v2 = (6, 0, 5, 1) and vz = (4, -7, 1, 3) in R* are linearly dependent or linearly independent. (b) Is it possible to express v, as a linear combination of v2 and v3? 2. Find the bases for the null space, row space and column space of the following matrix: 1 4 5 6 -2 4 -1 A = -1 -2 -1 5 -1 2 3 7 8. Hence mention the rank and nullity of A.

1. (a) Determine whether the following three vectors v, = (0, 3, 1,-1), v2 = (6, 0, 5, 1) and vz = (4, -7, 1, 3) in R* are linearly dependent or linearly independent. (b) Is it possible to express v, as a linear combination of v2 and v3? 2. Find the bases for the null space, row space and column space of the following matrix: 1 4 5 6 -2 4 -1 A = -1 -2 -1 5 -1 2 3 7 8. Hence mention the rank and nullity of A.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.6: The Matrix Of A Linear Transformation
Problem 7AEXP
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1. (a) Determine whether the following three vectors v, = (0, 3, 1,–1), V2 = (6, 0, 5, 1) and v3 = (4, -7, 1, 3) in R* are linearly dependent or linearly independent. (b) Is it possible to express v, as a linear combination of vz and v3? 2. Find the bases for the null space, row space and column space of the following matrix: 1 4 9. 3 A = -1 -2 4 -1 -1 -2 -1 2 7 8. Hence mention the rank and nullity of A. 3. Find the eigenvalues and the corresponding eigenvectors of the following matrix: 1 -3 3 A = 3 -5 3 \6 -6 4/ Find a matrix P that diagonalizes A. Hence calculate A7. 4. Find the steady-state vector q of the following regular transition matrix of a Markov chain by (a) iterative method (considering an initial vector) and (b) (0.90 0.15 0.10) linear system technique: P = 0.05 0.75 0.05 0.05 0.10 0.85/ 5. Determine whether the following three vectors v, = (1, 2, 3, 4), V2 = (0, 1, 0, –1) and vz = (1, 3, 3, 3) form a basis for R* or not ?
Transcribed Image Text:1. (a) Determine whether the following three vectors v, = (0, 3, 1,–1), V2 = (6, 0, 5, 1) and v3 = (4, -7, 1, 3) in R* are linearly dependent or linearly independent. (b) Is it possible to express v, as a linear combination of vz and v3? 2. Find the bases for the null space, row space and column space of the following matrix: 1 4 9. 3 A = -1 -2 4 -1 -1 -2 -1 2 7 8. Hence mention the rank and nullity of A. 3. Find the eigenvalues and the corresponding eigenvectors of the following matrix: 1 -3 3 A = 3 -5 3 \6 -6 4/ Find a matrix P that diagonalizes A. Hence calculate A7. 4. Find the steady-state vector q of the following regular transition matrix of a Markov chain by (a) iterative method (considering an initial vector) and (b) (0.90 0.15 0.10) linear system technique: P = 0.05 0.75 0.05 0.05 0.10 0.85/ 5. Determine whether the following three vectors v, = (1, 2, 3, 4), V2 = (0, 1, 0, –1) and vz = (1, 3, 3, 3) form a basis for R* or not ?
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