7) The vectors. A) True 13 False 17 -19 + -34 38 17 2 form a basis for n
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Linear Algebra: Basis of Vector Spaces
#### Problem Statement:
Determine whether the following vectors form a basis for \(\mathbb{R}^2\):
\[ \left\{ \begin{pmatrix}
17 \\
-19
\end{pmatrix},
\begin{pmatrix}
-34 \\
38
\end{pmatrix} \right\} \]
A) True
B) False
---
To determine if the given vectors form a basis for \(\mathbb{R}^2\), we need to check if they are linearly independent and span \(\mathbb{R}^2\). For two vectors to form a basis in \(\mathbb{R}^2\), their determinant must be non-zero.
Let's denote the vectors as \( \mathbf{v}_1 \) and \( \mathbf{v}_2 \):
\[ \mathbf{v}_1 = \begin{pmatrix} 17 \\ -19 \end{pmatrix} \]
\[ \mathbf{v}_2 = \begin{pmatrix} -34 \\ 38 \end{pmatrix} \]
Calculate the determinant of the matrix formed by \(\mathbf{v}_1\) and \(\mathbf{v}_2\):
\[ \text{det} \left( \begin{bmatrix} 17 & -34 \\ -19 & 38 \end{bmatrix} \right) = (17 \cdot 38) - (-19 \cdot -34) \]
\[ = 646 - 646 \]
\[ = 0 \]
Since the determinant is 0, the vectors \(\mathbf{v}_1\) and \(\mathbf{v}_2\) are linearly dependent and do not form a basis for \(\mathbb{R}^2\).
---
Therefore, the correct answer is:
B) False](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdc37c896-f9bf-421c-983e-bcaad18451c5%2F6ec40856-979a-4039-8015-c939283fad41%2Foyet58_processed.png&w=3840&q=75)
Transcribed Image Text:### Linear Algebra: Basis of Vector Spaces
#### Problem Statement:
Determine whether the following vectors form a basis for \(\mathbb{R}^2\):
\[ \left\{ \begin{pmatrix}
17 \\
-19
\end{pmatrix},
\begin{pmatrix}
-34 \\
38
\end{pmatrix} \right\} \]
A) True
B) False
---
To determine if the given vectors form a basis for \(\mathbb{R}^2\), we need to check if they are linearly independent and span \(\mathbb{R}^2\). For two vectors to form a basis in \(\mathbb{R}^2\), their determinant must be non-zero.
Let's denote the vectors as \( \mathbf{v}_1 \) and \( \mathbf{v}_2 \):
\[ \mathbf{v}_1 = \begin{pmatrix} 17 \\ -19 \end{pmatrix} \]
\[ \mathbf{v}_2 = \begin{pmatrix} -34 \\ 38 \end{pmatrix} \]
Calculate the determinant of the matrix formed by \(\mathbf{v}_1\) and \(\mathbf{v}_2\):
\[ \text{det} \left( \begin{bmatrix} 17 & -34 \\ -19 & 38 \end{bmatrix} \right) = (17 \cdot 38) - (-19 \cdot -34) \]
\[ = 646 - 646 \]
\[ = 0 \]
Since the determinant is 0, the vectors \(\mathbf{v}_1\) and \(\mathbf{v}_2\) are linearly dependent and do not form a basis for \(\mathbb{R}^2\).
---
Therefore, the correct answer is:
B) False
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