Determine if the set is a basis for R°. Justify your answer. 6 -4 1 1 - 4 Is the given set a basis for R3? O A. No, because these two vectors are linearly dependent. O B. Yes, because these vectors form the columns of an invertible 3 x3 matrix. By the invertible matrix theorem, the following statements are equivalent: A is an invertible matrix, the columns of A form a linearly independent set, and the columns of A span R". O C. No, because these vectors form a matrix with only 2 pivot columns. Therefore, these vectors form a basis for a two-dimensional subspace of R3. O D. Yes, because these two vectors are linearly independent.
Determine if the set is a basis for R°. Justify your answer. 6 -4 1 1 - 4 Is the given set a basis for R3? O A. No, because these two vectors are linearly dependent. O B. Yes, because these vectors form the columns of an invertible 3 x3 matrix. By the invertible matrix theorem, the following statements are equivalent: A is an invertible matrix, the columns of A form a linearly independent set, and the columns of A span R". O C. No, because these vectors form a matrix with only 2 pivot columns. Therefore, these vectors form a basis for a two-dimensional subspace of R3. O D. Yes, because these two vectors are linearly independent.
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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![### Determining Basis in \( \mathbb{R}^3 \)
#### Determine if the set is a basis for \( \mathbb{R}^3 \). Justify your answer.
\[
\left\{
\begin{pmatrix}
3 \\
-4 \\
1
\end{pmatrix},
\begin{pmatrix}
6 \\
1 \\
-4
\end{pmatrix}
\right\}
\]
#### Is the given set a basis for \( \mathbb{R}^3 \)?
- **A.** No, because these two vectors are linearly dependent.
- **B.** Yes, because these vectors form the columns of an invertible \( 3 \times 3 \) matrix. By the invertible matrix theorem, the following statements are equivalent: \( \mathbf{A} \) is an invertible matrix, the columns of \( \mathbf{A} \) form a linearly independent set, and the columns of \( \mathbf{A} \) span \( \mathbb{R}^n \).
- **C.** No, because these vectors form a matrix with only 2 pivot columns. Therefore, these vectors form a basis for a two-dimensional subspace of \( \mathbb{R}^3 \).
- **D.** Yes, because these two vectors are linearly independent.
To summarize, this question assesses whether the provided set of vectors is a basis for \( \mathbb{R}^3 \). A set of vectors is considered a basis for \( \mathbb{R}^3 \) if it satisfies two conditions:
1. The vectors must be linearly independent.
2. The vectors must span \( \mathbb{R}^3 \).
Since the set consists of only two vectors and \( \mathbb{R}^3 \) requires three linearly independent vectors to form a basis, the correct answer is **C.** No, because these vectors form a matrix with only 2 pivot columns. Therefore, these vectors form a basis for a two-dimensional subspace of \( \mathbb{R}^3 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc0867650-b445-489d-98f3-e27e64467d14%2Ff3e2e6f6-15aa-4206-bcbc-b7e578d28bee%2Fn9j3kqf_processed.png&w=3840&q=75)
Transcribed Image Text:### Determining Basis in \( \mathbb{R}^3 \)
#### Determine if the set is a basis for \( \mathbb{R}^3 \). Justify your answer.
\[
\left\{
\begin{pmatrix}
3 \\
-4 \\
1
\end{pmatrix},
\begin{pmatrix}
6 \\
1 \\
-4
\end{pmatrix}
\right\}
\]
#### Is the given set a basis for \( \mathbb{R}^3 \)?
- **A.** No, because these two vectors are linearly dependent.
- **B.** Yes, because these vectors form the columns of an invertible \( 3 \times 3 \) matrix. By the invertible matrix theorem, the following statements are equivalent: \( \mathbf{A} \) is an invertible matrix, the columns of \( \mathbf{A} \) form a linearly independent set, and the columns of \( \mathbf{A} \) span \( \mathbb{R}^n \).
- **C.** No, because these vectors form a matrix with only 2 pivot columns. Therefore, these vectors form a basis for a two-dimensional subspace of \( \mathbb{R}^3 \).
- **D.** Yes, because these two vectors are linearly independent.
To summarize, this question assesses whether the provided set of vectors is a basis for \( \mathbb{R}^3 \). A set of vectors is considered a basis for \( \mathbb{R}^3 \) if it satisfies two conditions:
1. The vectors must be linearly independent.
2. The vectors must span \( \mathbb{R}^3 \).
Since the set consists of only two vectors and \( \mathbb{R}^3 \) requires three linearly independent vectors to form a basis, the correct answer is **C.** No, because these vectors form a matrix with only 2 pivot columns. Therefore, these vectors form a basis for a two-dimensional subspace of \( \mathbb{R}^3 \).
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