Find a basis for the space spanned by the given vectors. 1 2 6. 4 - 1 -2 - 1 2 4 2 - 2 || -2 1 -7 - 3 A basis for the space spanned by the given vectors is (Use a comma to separate answers as needed.)
Find a basis for the space spanned by the given vectors. 1 2 6. 4 - 1 -2 - 1 2 4 2 - 2 || -2 1 -7 - 3 A basis for the space spanned by the given vectors is (Use a comma to separate answers as needed.)
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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Question
![**Find a Basis for the Space Spanned by the Given Vectors**
Given vectors:
\[
\begin{bmatrix}
1 \\
0 \\
0 \\
1
\end{bmatrix},
\begin{bmatrix}
2 \\
0 \\
0 \\
-2
\end{bmatrix},
\begin{bmatrix}
3 \\
-1 \\
2 \\
-2
\end{bmatrix},
\begin{bmatrix}
6 \\
-2 \\
4 \\
-7
\end{bmatrix},
\begin{bmatrix}
4 \\
-1 \\
2 \\
-3
\end{bmatrix}
\]
*A basis for the space spanned by the given vectors is ______. (Use a comma to separate answers as needed.)*
**Explanation:**
To find a basis for the space spanned by these vectors, one must determine which vectors are linearly independent. Typically, this involves forming a matrix with these vectors as columns and performing row reduction to find the pivot columns. The vectors corresponding to the pivot columns form a basis.
The process involves:
1. **Matrix Formation**: Arrange the vectors in a matrix.
2. **Row Reduction**: Use Gaussian elimination to reduce the matrix to row-echelon form.
3. **Identify Pivot Columns**: Determine the columns that contain leading ones.
4. **Select Basis Vectors**: Choose vectors corresponding to the pivot columns as the basis for the space.
This process results in a set of linearly independent vectors that span the same space as the original set.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faba31a81-9649-4cd9-8404-baa278a33196%2Fe7a7b7a5-0795-4dab-8aec-d9bc7816d146%2F4o1j5b2w_processed.png&w=3840&q=75)
Transcribed Image Text:**Find a Basis for the Space Spanned by the Given Vectors**
Given vectors:
\[
\begin{bmatrix}
1 \\
0 \\
0 \\
1
\end{bmatrix},
\begin{bmatrix}
2 \\
0 \\
0 \\
-2
\end{bmatrix},
\begin{bmatrix}
3 \\
-1 \\
2 \\
-2
\end{bmatrix},
\begin{bmatrix}
6 \\
-2 \\
4 \\
-7
\end{bmatrix},
\begin{bmatrix}
4 \\
-1 \\
2 \\
-3
\end{bmatrix}
\]
*A basis for the space spanned by the given vectors is ______. (Use a comma to separate answers as needed.)*
**Explanation:**
To find a basis for the space spanned by these vectors, one must determine which vectors are linearly independent. Typically, this involves forming a matrix with these vectors as columns and performing row reduction to find the pivot columns. The vectors corresponding to the pivot columns form a basis.
The process involves:
1. **Matrix Formation**: Arrange the vectors in a matrix.
2. **Row Reduction**: Use Gaussian elimination to reduce the matrix to row-echelon form.
3. **Identify Pivot Columns**: Determine the columns that contain leading ones.
4. **Select Basis Vectors**: Choose vectors corresponding to the pivot columns as the basis for the space.
This process results in a set of linearly independent vectors that span the same space as the original set.
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