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Find the column space of the 10 × 20 matrix A if the system Ax = b is solvable for every b. Justify your answer.

Find the column space of the 10 × 20 matrix A if the system Ax = b is solvable for every b. Justify your answer.

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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## Problem Statement **Find the column space of the \(10 \times 20\) matrix \(A\) if the system \(A\mathbf{x} = \mathbf{b}\) is solvable for every \(\mathbf{b}\).** ### Explanation 1. **Matrix Dimensions and Column Space:** - A matrix with dimensions \(10 \times 20\) means it has 10 rows and 20 columns. - The column space of a matrix is the set of all possible linear combinations of its column vectors. 2. **System Solvability:** - If the system \(A\mathbf{x} = \mathbf{b}\) is solvable for every vector \(\mathbf{b}\) in \(\mathbb{R}^{10}\), then the transformation represented by the matrix \(A\) must cover the entire space \(\mathbb{R}^{10}\). 3. **Full Column Rank:** - For \(A\) to be solvable for every \(\mathbf{b}\), \(A\) must have full row rank. Therefore, the column space of \(A\) must be all of \(\mathbb{R}^{10}\). 4. **Conclusion:** - The column space of \(A\) is \(\mathbb{R}^{10}\). ### Justification Since the system is solvable for every \(\mathbf{b}\) in \(\mathbb{R}^{10}\), \(A\) covers the entire \(\mathbb{R}^{10}\) space, which implies that the column space of \(A\) spans \(\mathbb{R}^{10}\).
Transcribed Image Text:## Problem Statement **Find the column space of the \(10 \times 20\) matrix \(A\) if the system \(A\mathbf{x} = \mathbf{b}\) is solvable for every \(\mathbf{b}\).** ### Explanation 1. **Matrix Dimensions and Column Space:** - A matrix with dimensions \(10 \times 20\) means it has 10 rows and 20 columns. - The column space of a matrix is the set of all possible linear combinations of its column vectors. 2. **System Solvability:** - If the system \(A\mathbf{x} = \mathbf{b}\) is solvable for every vector \(\mathbf{b}\) in \(\mathbb{R}^{10}\), then the transformation represented by the matrix \(A\) must cover the entire space \(\mathbb{R}^{10}\). 3. **Full Column Rank:** - For \(A\) to be solvable for every \(\mathbf{b}\), \(A\) must have full row rank. Therefore, the column space of \(A\) must be all of \(\mathbb{R}^{10}\). 4. **Conclusion:** - The column space of \(A\) is \(\mathbb{R}^{10}\). ### Justification Since the system is solvable for every \(\mathbf{b}\) in \(\mathbb{R}^{10}\), \(A\) covers the entire \(\mathbb{R}^{10}\) space, which implies that the column space of \(A\) spans \(\mathbb{R}^{10}\).
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