Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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### Linear Algebra: Determining a Basis in ℝ³
**Problem Statement:**
The set \( S = \left\{
\begin{pmatrix}
0 \\ -10 \\ -6
\end{pmatrix},
\begin{pmatrix}
-1 \\ 6 \\ k
\end{pmatrix},
\begin{pmatrix}
k \\ 6 \\ -8
\end{pmatrix}
\right\} \) is a basis for \(\mathbb{R}^3\) if and only if:
\[ k \neq \]
\[ \underline{\hspace{2cm}} \]
---
**Instructions to Answer the Question:**
To determine the value of \( k \) for which the set \( S \) is a basis for \(\mathbb{R}^3\), you need to ensure that these three vectors are linearly independent. Therefore, compute the determinant of the matrix formed by these vectors interpreted as columns:
\[
\text{Matrix} \: A =
\begin{pmatrix}
0 & -1 & k \\
-10 & 6 & 6 \\
-6 & k & -8
\end{pmatrix}
\]
Set the determinant of the matrix \( A \) to be non-zero.
**Next Steps:**
1. Compute the determinant of matrix \( A \).
2. Find the condition on \( k \) where the determinant is non-zero.
3. Enter the value(s) of \( k \) that makes the set \( S \) a basis for \(\mathbb{R}^3\).
---
**Problems to Solve Next**
Click on "Next Question" to proceed to subsequent problems involving basis and linear independence in vector spaces.
\[ > \text{ Next Question } \]
---](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F27b1de37-aa3c-4b4b-ac83-7a3d52bec671%2F39a66b8d-6a9f-4c47-85d0-0164a6ec14c0%2Fq532j8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:---
### Linear Algebra: Determining a Basis in ℝ³
**Problem Statement:**
The set \( S = \left\{
\begin{pmatrix}
0 \\ -10 \\ -6
\end{pmatrix},
\begin{pmatrix}
-1 \\ 6 \\ k
\end{pmatrix},
\begin{pmatrix}
k \\ 6 \\ -8
\end{pmatrix}
\right\} \) is a basis for \(\mathbb{R}^3\) if and only if:
\[ k \neq \]
\[ \underline{\hspace{2cm}} \]
---
**Instructions to Answer the Question:**
To determine the value of \( k \) for which the set \( S \) is a basis for \(\mathbb{R}^3\), you need to ensure that these three vectors are linearly independent. Therefore, compute the determinant of the matrix formed by these vectors interpreted as columns:
\[
\text{Matrix} \: A =
\begin{pmatrix}
0 & -1 & k \\
-10 & 6 & 6 \\
-6 & k & -8
\end{pmatrix}
\]
Set the determinant of the matrix \( A \) to be non-zero.
**Next Steps:**
1. Compute the determinant of matrix \( A \).
2. Find the condition on \( k \) where the determinant is non-zero.
3. Enter the value(s) of \( k \) that makes the set \( S \) a basis for \(\mathbb{R}^3\).
---
**Problems to Solve Next**
Click on "Next Question" to proceed to subsequent problems involving basis and linear independence in vector spaces.
\[ > \text{ Next Question } \]
---
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