Find a basis for the column space and the rank of the matrix. -2 -4 -2 1 -3 14 7 -6 -6 4 2 -3 -2 4 2 -2 (a) a basis for the column space
Find a basis for the column space and the rank of the matrix. -2 -4 -2 1 -3 14 7 -6 -6 4 2 -3 -2 4 2 -2 (a) a basis for the column space
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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![### Finding a Basis for the Column Space and the Rank of the Matrix
Consider the matrix:
\[
\begin{bmatrix}
-2 & -4 & -2 & 1 \\
-3 & 14 & 7 & -6 \\
-6 & 4 & 2 & -3 \\
-2 & 4 & 2 & -2
\end{bmatrix}
\]
#### Task:
1. **Find a basis for the column space.**
2. **Determine the rank of the matrix.**
#### Explanation:
To find the basis for the column space, identify the set of linearly independent columns. This involves manipulating the matrix to its row-echelon form and selecting columns that correspond to the leading entries (pivots).
#### Diagram Description:
- The diagram represents a schematic process of selecting columns from the matrix.
- Two groups of boxed columns are shown between curly braces.
- Arrows indicate the direction of selection, suggesting which columns are pivotal.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8268c271-8991-49a2-aed9-a02bde5bd8ab%2Ff5338dc2-78cd-49ce-8076-8cbebba346e2%2Fs4n5dqp_processed.png&w=3840&q=75)
Transcribed Image Text:### Finding a Basis for the Column Space and the Rank of the Matrix
Consider the matrix:
\[
\begin{bmatrix}
-2 & -4 & -2 & 1 \\
-3 & 14 & 7 & -6 \\
-6 & 4 & 2 & -3 \\
-2 & 4 & 2 & -2
\end{bmatrix}
\]
#### Task:
1. **Find a basis for the column space.**
2. **Determine the rank of the matrix.**
#### Explanation:
To find the basis for the column space, identify the set of linearly independent columns. This involves manipulating the matrix to its row-echelon form and selecting columns that correspond to the leading entries (pivots).
#### Diagram Description:
- The diagram represents a schematic process of selecting columns from the matrix.
- Two groups of boxed columns are shown between curly braces.
- Arrows indicate the direction of selection, suggesting which columns are pivotal.
![**Problem Statement:**
Find a basis for the subspace of \( \mathbb{R}^4 \) spanned by \( S \).
**Set \( S \):**
\[
S = \{ (5, 9, -5, 53), \, (-2, 5, 2, -5), \, (8, -2, -8, 17), \, (0, -2, 0, 15) \}
\]
**Explanation of the Diagram:**
The diagram visually represents arranging the vectors from set \( S \) into a matrix form. The vectors, written as row vectors, are positioned inside a large bracket system indicating a matrix. The vertical and horizontal double arrows suggest operations or transformations such as row reduction might follow to find the basis.
The vectors as seen in matrix form (not filled in):
\[
\begin{bmatrix}
\boxed{\phantom{5\, 9\, -5\, 53}} \\
\boxed{\phantom{-2\, 5\, 2\, -5}} \\
\boxed{\phantom{8\, -2\, -8\, 17}} \\
\boxed{\phantom{0\, -2\, 0\, 15}}
\end{bmatrix}
\]
Further steps involve using operations like row reduction to bring the matrix into row-echelon form, from which you can determine the independent vectors forming the basis for the span of \( S \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8268c271-8991-49a2-aed9-a02bde5bd8ab%2Ff5338dc2-78cd-49ce-8076-8cbebba346e2%2F2h51f98_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find a basis for the subspace of \( \mathbb{R}^4 \) spanned by \( S \).
**Set \( S \):**
\[
S = \{ (5, 9, -5, 53), \, (-2, 5, 2, -5), \, (8, -2, -8, 17), \, (0, -2, 0, 15) \}
\]
**Explanation of the Diagram:**
The diagram visually represents arranging the vectors from set \( S \) into a matrix form. The vectors, written as row vectors, are positioned inside a large bracket system indicating a matrix. The vertical and horizontal double arrows suggest operations or transformations such as row reduction might follow to find the basis.
The vectors as seen in matrix form (not filled in):
\[
\begin{bmatrix}
\boxed{\phantom{5\, 9\, -5\, 53}} \\
\boxed{\phantom{-2\, 5\, 2\, -5}} \\
\boxed{\phantom{8\, -2\, -8\, 17}} \\
\boxed{\phantom{0\, -2\, 0\, 15}}
\end{bmatrix}
\]
Further steps involve using operations like row reduction to bring the matrix into row-echelon form, from which you can determine the independent vectors forming the basis for the span of \( S \).
Expert Solution
Recall:
The number of vectors in the bbasis for column space is the rank of the matrix.
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