2. Determine if the following sets of vectors are linearly independent. Then determine if they form a basis for the specified space. Explain your reasoning. а. ,R³ b. -個 С.
2. Determine if the following sets of vectors are linearly independent. Then determine if they form a basis for the specified space. Explain your reasoning. а. ,R³ b. -個 С.
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
![**Problem 2: Linear Independence and Basis of Vector Spaces**
**Task:**
Determine if the following sets of vectors are linearly independent. Then determine if they form a basis for the specified space. Explain your reasoning.
**a. Vectors in \( \mathbb{R}^3 \):**
\[
\left\{
\begin{bmatrix}
1 \\
6 \\
2
\end{bmatrix},
\begin{bmatrix}
0 \\
1 \\
1
\end{bmatrix}
\right\}
\]
**b. Vectors in \( \mathbb{R}^2 \):**
\[
\left\{
\begin{bmatrix}
1 \\
3
\end{bmatrix},
\begin{bmatrix}
2 \\
2
\end{bmatrix},
\begin{bmatrix}
-1 \\
1
\end{bmatrix}
\right\}
\]
**c. Vectors in \( \mathbb{R}^4 \):**
\[
\left\{
\begin{bmatrix}
1 \\
2 \\
1 \\
0
\end{bmatrix},
\begin{bmatrix}
-1 \\
1 \\
3 \\
1
\end{bmatrix},
\begin{bmatrix}
5 \\
0 \\
1 \\
2
\end{bmatrix},
\begin{bmatrix}
3 \\
-1 \\
3 \\
3
\end{bmatrix}
\right\}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F410816ed-4be2-4291-b0fb-4c8f4b2206ab%2Fa673a96d-2845-42e3-b647-207e3793a6dc%2Fdk51s9_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 2: Linear Independence and Basis of Vector Spaces**
**Task:**
Determine if the following sets of vectors are linearly independent. Then determine if they form a basis for the specified space. Explain your reasoning.
**a. Vectors in \( \mathbb{R}^3 \):**
\[
\left\{
\begin{bmatrix}
1 \\
6 \\
2
\end{bmatrix},
\begin{bmatrix}
0 \\
1 \\
1
\end{bmatrix}
\right\}
\]
**b. Vectors in \( \mathbb{R}^2 \):**
\[
\left\{
\begin{bmatrix}
1 \\
3
\end{bmatrix},
\begin{bmatrix}
2 \\
2
\end{bmatrix},
\begin{bmatrix}
-1 \\
1
\end{bmatrix}
\right\}
\]
**c. Vectors in \( \mathbb{R}^4 \):**
\[
\left\{
\begin{bmatrix}
1 \\
2 \\
1 \\
0
\end{bmatrix},
\begin{bmatrix}
-1 \\
1 \\
3 \\
1
\end{bmatrix},
\begin{bmatrix}
5 \\
0 \\
1 \\
2
\end{bmatrix},
\begin{bmatrix}
3 \\
-1 \\
3 \\
3
\end{bmatrix}
\right\}
\]
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