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9. Given the LU decompositions of the following matrices A, give bases for R(A), N(A), C(A), and N(AT). (Do not multiply out!) Check dimensions and orthogonality. 1 a. A = 2 2 2 4 1 -1 | 1 1 1 2 1 *b. A = 1 1 2 -1 -2 0 1 0 0 0

9. Given the LU decompositions of the following matrices A, give bases for R(A), N(A), C(A), and N(AT). (Do not multiply out!) Check dimensions and orthogonality. 1 a. A = 2 2 2 4 1 -1 | 1 1 1 2 1 *b. A = 1 1 2 -1 -2 0 1 0 0 0

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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linear 3.4 Q9 sub a and b 

9. Given the LU decompositions of the following matrices \( A \), give bases for \( R(A) \), \( N(A) \), \( C(A) \), and \( N(A^T) \). (Do not multiply out!) Check dimensions and orthogonality. a. \( A = \begin{bmatrix} 1 & 0 \\ 2 & 1 \end{bmatrix} \begin{bmatrix} 2 & 2 & 4 \\ 0 & 1 & -1 \end{bmatrix} \) *b. \( A = \begin{bmatrix} 1 & 0 & 0 \\ 1 & 1 & 0 \\ -2 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 1 & 2 & 1 \\ 0 & 0 & 2 & -1 \\ 0 & 0 & 0 & 0 \end{bmatrix} \) c. \( A = \begin{bmatrix} 1 & 0 & 0 \\ -1 & 1 & 0 \\ 0 & 2 & 1 \\ 3 & 0 & 1 \end{bmatrix} \begin{bmatrix} 2 & 4 & -2 \\ 0 & 0 & 0 \\ 0 & 0 & 3 \\ 0 & 0 & 0 \end{bmatrix} \) In each case, consider the dimensions of the matrix and the orthogonality of the vectors within the provided matrices to establish the bases for the row space \( R(A) \), null space \( N(A) \), column space \( C(A) \), and the null space of the transpose \( N(A^T) \).
Transcribed Image Text:9. Given the LU decompositions of the following matrices \( A \), give bases for \( R(A) \), \( N(A) \), \( C(A) \), and \( N(A^T) \). (Do not multiply out!) Check dimensions and orthogonality. a. \( A = \begin{bmatrix} 1 & 0 \\ 2 & 1 \end{bmatrix} \begin{bmatrix} 2 & 2 & 4 \\ 0 & 1 & -1 \end{bmatrix} \) *b. \( A = \begin{bmatrix} 1 & 0 & 0 \\ 1 & 1 & 0 \\ -2 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 1 & 2 & 1 \\ 0 & 0 & 2 & -1 \\ 0 & 0 & 0 & 0 \end{bmatrix} \) c. \( A = \begin{bmatrix} 1 & 0 & 0 \\ -1 & 1 & 0 \\ 0 & 2 & 1 \\ 3 & 0 & 1 \end{bmatrix} \begin{bmatrix} 2 & 4 & -2 \\ 0 & 0 & 0 \\ 0 & 0 & 3 \\ 0 & 0 & 0 \end{bmatrix} \) In each case, consider the dimensions of the matrix and the orthogonality of the vectors within the provided matrices to establish the bases for the row space \( R(A) \), null space \( N(A) \), column space \( C(A) \), and the null space of the transpose \( N(A^T) \).
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