Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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#4 4.3
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![### Determine Basis and Linear Independence
Determine if the set of vectors shown to the right is a basis for \(\mathbb{R}^3\). If the set of vectors is not a basis, determine whether it is linearly independent and whether the set spans \(\mathbb{R}^3\).
\[
\left\{
\begin{bmatrix}
3 \\
-1 \\
1
\end{bmatrix},
\begin{bmatrix}
2 \\
-4 \\
2
\end{bmatrix},
\begin{bmatrix}
-9 \\
6 \\
3
\end{bmatrix}
\right\}
\]
---
### Multiple Choice Question
Which of the following describe the set? Select all that apply.
- [ ] A. The set is a basis for \(\mathbb{R}^3\).
- [ ] B. The set is linearly independent.
- [ ] C. The set spans \(\mathbb{R}^3\).
- [ ] D. None of the above
---
To solve the question, analyze if the given vectors are linearly independent and if they span the vector space \(\mathbb{R}^3\). A set of three vectors is a basis for \(\mathbb{R}^3\) if it is linearly independent and spans \(\mathbb{R}^3\).](https://content.bartleby.com/qna-images/question/ff5f1b79-f8dc-48a5-a6af-e3b161b5a7d5/0114d10f-2631-4c4e-982b-a817513dbd22/hoq2l4_processed.png)
Transcribed Image Text:### Determine Basis and Linear Independence
Determine if the set of vectors shown to the right is a basis for \(\mathbb{R}^3\). If the set of vectors is not a basis, determine whether it is linearly independent and whether the set spans \(\mathbb{R}^3\).
\[
\left\{
\begin{bmatrix}
3 \\
-1 \\
1
\end{bmatrix},
\begin{bmatrix}
2 \\
-4 \\
2
\end{bmatrix},
\begin{bmatrix}
-9 \\
6 \\
3
\end{bmatrix}
\right\}
\]
---
### Multiple Choice Question
Which of the following describe the set? Select all that apply.
- [ ] A. The set is a basis for \(\mathbb{R}^3\).
- [ ] B. The set is linearly independent.
- [ ] C. The set spans \(\mathbb{R}^3\).
- [ ] D. None of the above
---
To solve the question, analyze if the given vectors are linearly independent and if they span the vector space \(\mathbb{R}^3\). A set of three vectors is a basis for \(\mathbb{R}^3\) if it is linearly independent and spans \(\mathbb{R}^3\).
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