For what values of b does the following set of vectors form a basis? 3 (@··} 12 15
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:**
For what values of \( b \) does the following set of vectors form a basis?
\[
\left\{
\begin{bmatrix}
1 \\
4 \\
5
\end{bmatrix},
\begin{bmatrix}
3 \\
b \\
-5
\end{bmatrix},
\begin{bmatrix}
3 \\
12 \\
15
\end{bmatrix}
\right\}
\]
**Explanation:**
To determine the values of \( b \) for which the set of vectors forms a basis, we need to check if the vectors are linearly independent. This can be checked by calculating the determinant of the matrix formed by these vectors and ensuring it is non-zero.
The vectors form the columns of the following matrix:
\[
\begin{bmatrix}
1 & 3 & 3 \\
4 & b & 12 \\
5 & -5 & 15
\end{bmatrix}
\]
The determinant of this 3x3 matrix must be non-zero for the vectors to form a basis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcdd779ee-39b4-455c-9a15-b1302cac6e33%2Fe8819b5f-0004-4e56-aee1-eb09b32e42a0%2Fr3nwhzh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Question:**
For what values of \( b \) does the following set of vectors form a basis?
\[
\left\{
\begin{bmatrix}
1 \\
4 \\
5
\end{bmatrix},
\begin{bmatrix}
3 \\
b \\
-5
\end{bmatrix},
\begin{bmatrix}
3 \\
12 \\
15
\end{bmatrix}
\right\}
\]
**Explanation:**
To determine the values of \( b \) for which the set of vectors forms a basis, we need to check if the vectors are linearly independent. This can be checked by calculating the determinant of the matrix formed by these vectors and ensuring it is non-zero.
The vectors form the columns of the following matrix:
\[
\begin{bmatrix}
1 & 3 & 3 \\
4 & b & 12 \\
5 & -5 & 15
\end{bmatrix}
\]
The determinant of this 3x3 matrix must be non-zero for the vectors to form a basis.
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