**Problem:**Find a basis for the vector space of all $3 \times 3$ diagonal matrices.**Options:**1. \[ \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix} \]2. \[ \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix} \]3. \[ \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix} \]4. \[ \begin{pmatrix}

First we need to to find basis of vector space:

Find a basis for the vector space of all 3 x 3 diagonal matrices. 0 0 1 0 0 1 1 0 0] [o 1 0 10 0 0 1 0 1 0 0 0 1 0 0 0 1 10 0 0 1 0 0 0 1 0 1 0 0 10,0 1 o 0 0 1 0 1 0 10 0 0 0 0 [0 0 0 0 0 0.0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 o o] 0 0 0 o o|.1 0 o 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 o|. 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 What is the dimension of this vector space?

Find a basis for the vector space of all 3 x 3 diagonal matrices. 0 0 1 0 0 1 1 0 0] [o 1 0 10 0 0 1 0 1 0 0 0 1 0 0 0 1 10 0 0 1 0 0 0 1 0 1 0 0 10,0 1 o 0 0 1 0 1 0 10 0 0 0 0 [0 0 0 0 0 0.0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 o o] 0 0 0 o o|.1 0 o 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 o|. 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 What is the dimension of this vector space?

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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Concept explainers

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

**Problem:**

Find a basis for the vector space of all $3 \times 3$ diagonal matrices.

**Options:**

1.
\[
\begin{pmatrix}
1 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix},
\begin{pmatrix}
0 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 0
\end{pmatrix},
\begin{pmatrix}
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 1
\end{pmatrix}
\]

2.
\[
\begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 0
\end{pmatrix},
\begin{pmatrix}
0 & 0 & 0 \\
1 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix},
\begin{pmatrix}
0 & 0 & 1 \\
0 & 0 & 0 \\
0 & 1 & 0
\end{pmatrix}
\]

3.
\[
\begin{pmatrix}
1 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix},
\begin{pmatrix}
0 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 0
\end{pmatrix},
\begin{pmatrix}
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 1
\end{pmatrix}
\]

4.
\[
\begin{pmatrix}

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