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(g) The pivot columns of a matrix are always linearly independent. (h) If A is m x n and rank A = m, then the linear transformation x→ Ax is onto.

(g) The pivot columns of a matrix are always linearly independent. (h) If A is m x n and rank A = m, then the linear transformation x→ Ax is onto.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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(g) The pivot columns of a matrix are always linearly independent. (h) If A is m x n and rank A = m, then the linear transformation x→ Ax is onto. 4. The matrix A and its RREF are given below: [204 0 3 1 2 2 0 1 0 2 1 A = 1 RREF(A) 10 200 01 02 0 00001 00000 0 -10-2 (a) Find a basis for Col A and determine its dimension. (b) Find a basis for Nul A and determine its dimension. (c) Find a basis for Row A and determine its dimension. (d) How many pivot columns are in a row echelon form of AT? Explain. 5. Let A be the matrix 1 123 34 k (a) Find the values of k (if any) for which A is invertible. (b) Find the values of k (if any) for which A has rank 3. (c) Same for rank 2. (d) Same for rank 1.
Transcribed Image Text:(g) The pivot columns of a matrix are always linearly independent. (h) If A is m x n and rank A = m, then the linear transformation x→ Ax is onto. 4. The matrix A and its RREF are given below: [204 0 3 1 2 2 0 1 0 2 1 A = 1 RREF(A) 10 200 01 02 0 00001 00000 0 -10-2 (a) Find a basis for Col A and determine its dimension. (b) Find a basis for Nul A and determine its dimension. (c) Find a basis for Row A and determine its dimension. (d) How many pivot columns are in a row echelon form of AT? Explain. 5. Let A be the matrix 1 123 34 k (a) Find the values of k (if any) for which A is invertible. (b) Find the values of k (if any) for which A has rank 3. (c) Same for rank 2. (d) Same for rank 1.
*1. Let W be the subspace of R5 spanned by the vectors B Find a basis for W. Give the dimension of W. -3 b₁ = 2 *2. Consider B = {b₁,b₂}, a basis for R² where - (²). b₂ = (3) 0 (a) Find the change of basis matrix that converts vectors given with respect to B into vectors given with respect to the standard basis. Give your matrix explicitly. (b) Find the change of basis matrix that converts vectors given with respect to the standard basis into vectors given with respect to B. Give your matrix explicitly. 3. True or False? (a) There is a 5 x 3 matrix whose column space is 1-dimensional. (b) If A is an m x n matrix, then rank(A) + nullity(A) = m. (c) If A is an n x n matrix and the columns of A span R", then A is invertible. (d) If B is a 2 x 2 matrix and B² = I₂, then B = I2. (e) The kernel of the function T: R² R² given by T + (²) - (² + 1) ₁ = is {0}. (f) There is a matrix that has a 3-dimensional column space and a 2-dimensional row space.
Transcribed Image Text:*1. Let W be the subspace of R5 spanned by the vectors B Find a basis for W. Give the dimension of W. -3 b₁ = 2 *2. Consider B = {b₁,b₂}, a basis for R² where - (²). b₂ = (3) 0 (a) Find the change of basis matrix that converts vectors given with respect to B into vectors given with respect to the standard basis. Give your matrix explicitly. (b) Find the change of basis matrix that converts vectors given with respect to the standard basis into vectors given with respect to B. Give your matrix explicitly. 3. True or False? (a) There is a 5 x 3 matrix whose column space is 1-dimensional. (b) If A is an m x n matrix, then rank(A) + nullity(A) = m. (c) If A is an n x n matrix and the columns of A span R", then A is invertible. (d) If B is a 2 x 2 matrix and B² = I₂, then B = I2. (e) The kernel of the function T: R² R² given by T + (²) - (² + 1) ₁ = is {0}. (f) There is a matrix that has a 3-dimensional column space and a 2-dimensional row space.
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