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A matrix A and an echelon form of A are shown below. Find a basis for Col A and a basis for Nul A. A = 57 3 - 26 4 64-14 23 2-12 Find a basis for Col A. 1 2 3 - 10 014 -8 000 0

A matrix A and an echelon form of A are shown below. Find a basis for Col A and a basis for Nul A. A = 57 3 - 26 4 64-14 23 2-12 Find a basis for Col A. 1 2 3 - 10 014 -8 000 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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### Matrix and Echelon Form A matrix \( A \) and an echelon form of \( A \) are shown below. Find a basis for \( \text{Col } A \) and a basis for \( \text{Nul } A \). \[ A = \begin{bmatrix} 5 & 7 & 3 & -26 \\ 6 & 4 & -14 & 4 \\ 2 & 3 & 2 & -12 \end{bmatrix} \sim \begin{bmatrix} 1 & 2 & 3 & -10 \\ 0 & 1 & 4 & -8 \\ 0 & 0 & 0 & 0 \end{bmatrix} \] --- ### Instructions **Find a basis for \( \text{Col } A \).** - Identify the pivot columns in the echelon form of the matrix. - Use the corresponding columns from the original matrix \( A \) to form a basis for \( \text{Col } A \). **Identifying Pivot Columns:** 1. The first column [1, 0, 0] corresponds to a pivot column. 2. The second column [2, 1, 0] corresponds to a pivot column. Use the columns from the original matrix \( A \) that correspond to the pivot columns: - Column 1 from matrix \( A \): \([5, 6, 2]\) - Column 2 from matrix \( A \): \([7, 4, 3]\) Thus, the basis for \( \text{Col } A \) is \(\{ [5, 6, 2], [7, 4, 3] \}\).
Transcribed Image Text:### Matrix and Echelon Form A matrix \( A \) and an echelon form of \( A \) are shown below. Find a basis for \( \text{Col } A \) and a basis for \( \text{Nul } A \). \[ A = \begin{bmatrix} 5 & 7 & 3 & -26 \\ 6 & 4 & -14 & 4 \\ 2 & 3 & 2 & -12 \end{bmatrix} \sim \begin{bmatrix} 1 & 2 & 3 & -10 \\ 0 & 1 & 4 & -8 \\ 0 & 0 & 0 & 0 \end{bmatrix} \] --- ### Instructions **Find a basis for \( \text{Col } A \).** - Identify the pivot columns in the echelon form of the matrix. - Use the corresponding columns from the original matrix \( A \) to form a basis for \( \text{Col } A \). **Identifying Pivot Columns:** 1. The first column [1, 0, 0] corresponds to a pivot column. 2. The second column [2, 1, 0] corresponds to a pivot column. Use the columns from the original matrix \( A \) that correspond to the pivot columns: - Column 1 from matrix \( A \): \([5, 6, 2]\) - Column 2 from matrix \( A \): \([7, 4, 3]\) Thus, the basis for \( \text{Col } A \) is \(\{ [5, 6, 2], [7, 4, 3] \}\).
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