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36. Suppose a 5 x 6 matrix A has four pivot columns. Wha nullity A? Is Col A = R4? Why or why not?

36. Suppose a 5 x 6 matrix A has four pivot columns. Wha nullity A? Is Col A = R4? Why or why not?

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 36:** Suppose a \(5 \times 6\) matrix \(A\) has four pivot columns. What is the nullity \(A\)? Is \(\text{Col} \, A = \mathbb{R}^4\)? Why or why not? **Explanations:** 1. **Matrix and Dimensions:** - The matrix \(A\) is a \(5 \times 6\) matrix, meaning it has 5 rows and 6 columns. 2. **Pivot Columns:** - The matrix has four pivot columns, which are columns of the matrix that contain leading entries (or "pivots") in the row-reduced form of the matrix. 3. **Nullity of A:** - Nullity is defined as the number of columns minus the rank of the matrix. - Rank is the number of pivot columns, which is given as 4. - Thus, nullity of \(A = \text{Number of columns} - \text{Rank} = 6 - 4 = 2\). 4. **Column Space:** - \(\text{Col} \, A\) refers to the column space of \(A\), which is the set of all possible linear combinations of its column vectors. - Since \(A\) is a \(5 \times 6\) matrix and has 4 pivot columns, the dimension of \(\text{Col} \, A\) is 4. - However, \(\text{Col} \, A\) consists of vectors that are in \(\mathbb{R}^5\), not \(\mathbb{R}^4\), because the columns of \(A\) are 5-dimensional vectors. Therefore, \(\text{Col} \, A\) cannot be equal to \(\mathbb{R}^4\). The column space \(\text{Col} \, A\) is actually a 4-dimensional subspace of \(\mathbb{R}^5\).
Transcribed Image Text:**Problem 36:** Suppose a \(5 \times 6\) matrix \(A\) has four pivot columns. What is the nullity \(A\)? Is \(\text{Col} \, A = \mathbb{R}^4\)? Why or why not? **Explanations:** 1. **Matrix and Dimensions:** - The matrix \(A\) is a \(5 \times 6\) matrix, meaning it has 5 rows and 6 columns. 2. **Pivot Columns:** - The matrix has four pivot columns, which are columns of the matrix that contain leading entries (or "pivots") in the row-reduced form of the matrix. 3. **Nullity of A:** - Nullity is defined as the number of columns minus the rank of the matrix. - Rank is the number of pivot columns, which is given as 4. - Thus, nullity of \(A = \text{Number of columns} - \text{Rank} = 6 - 4 = 2\). 4. **Column Space:** - \(\text{Col} \, A\) refers to the column space of \(A\), which is the set of all possible linear combinations of its column vectors. - Since \(A\) is a \(5 \times 6\) matrix and has 4 pivot columns, the dimension of \(\text{Col} \, A\) is 4. - However, \(\text{Col} \, A\) consists of vectors that are in \(\mathbb{R}^5\), not \(\mathbb{R}^4\), because the columns of \(A\) are 5-dimensional vectors. Therefore, \(\text{Col} \, A\) cannot be equal to \(\mathbb{R}^4\). The column space \(\text{Col} \, A\) is actually a 4-dimensional subspace of \(\mathbb{R}^5\).
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