**25.** Let $ A = \begin{bmatrix}1 & 0 & -4 \\0 & 3 & -2 \\-2 & 6 & 3 \end{bmatrix} $ and $ \mathbf{b} = \begin{bmatrix}4 \\1 \\-4 \end{bmatrix} $. Denote the columns of $ A $ by $ \mathbf{a_1}, \mathbf{a_2}, \mathbf{a_3} $, and let $ W = \text{Span} \{ \mathbf{a_1}, \mathbf{a_2}, \mathbf{a_3} \} $. ### Linear Algebra Exercises1. **Is $ \mathbf{b} $ in $\{ \mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3 \}$? How many vectors are in $\{ \mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3 \}$?** 2. **Is $ \mathbf{b} $ in $ W $? How many vectors are in $ W $?** 3. **Show that $\mathbf{a}_1$ is in $ W $. [**Hint:** Row operations are unnecessary.]**In these exercises, you are asked to analyze the relationship between different vectors and vector spaces. For these problems, consider:- $\mathbf{b}$, $\mathbf{a}_1$, $\mathbf{a}_2$, and $\mathbf{a}_3$ are vectors in a given vector space.- $W$ represents a subspace or another vector space that we are interested in.### Detailed Explanations:- **Question (a):** - Determine whether the vector $ \mathbf{b} $ is an element of the set $ \{ \mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3 \} $. - Count the number of vectors included in the set $ \{ \mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3 \} $.- **Question (b):** - Determine whether the vector $ \mathbf{b} $ is an element of the vector space $ W $. - Count the number of vectors included in the vector space $ W $.- **Question (c):** - Prove that the vector $\mathbf{a}_1$ is an element of the vector space $ W $. - Note that row operations are not needed for this proof; try to use other properties of the vectors or the definition of the vector space $ W $.

The question is related to vector spaces. The detailed explanation is given below.

Answered: a. Is b in {a1, a2, a3}? How many…

a. Is b in {a1, a2, a3}? How many vectors are in {a¡, a2, a3}? b. Is b in W? How many vectors are in W? c. Show that a¡ is in W. [Hint: Row operations are unnec- essary.]

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

**25.** Let $ A = \begin{bmatrix}
1 & 0 & -4 \\
0 & 3 & -2 \\
-2 & 6 & 3
\end{bmatrix} $ and $ \mathbf{b} = \begin{bmatrix}
4 \\
1 \\
-4
\end{bmatrix} $. Denote the columns of $ A $ by $ \mathbf{a_1}, \mathbf{a_2}, \mathbf{a_3} $, and let $ W = \text{Span} \{ \mathbf{a_1}, \mathbf{a_2}, \mathbf{a_3} \} $.

### Linear Algebra Exercises

1. **Is $ \mathbf{b} $ in $\{ \mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3 \}$? How many vectors are in $\{ \mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3 \}$?**

2. **Is $ \mathbf{b} $ in $ W $? How many vectors are in $ W $?**

3. **Show that $\mathbf{a}_1$ is in $ W $. [**Hint:** Row operations are unnecessary.]**

In these exercises, you are asked to analyze the relationship between different vectors and vector spaces. For these problems, consider:
- $\mathbf{b}$, $\mathbf{a}_1$, $\mathbf{a}_2$, and $\mathbf{a}_3$ are vectors in a given vector space.
- $W$ represents a subspace or another vector space that we are interested in.

### Detailed Explanations:

- **Question (a):**
- Determine whether the vector $ \mathbf{b} $ is an element of the set $ \{ \mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3 \} $.
- Count the number of vectors included in the set $ \{ \mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3 \} $.

- **Question (b):**
- Determine whether the vector $ \mathbf{b} $ is an element of the vector space $ W $.
- Count the number of vectors included in the vector space $ W $.

- **Question (c):**
- Prove that the vector $\mathbf{a}_1$ is an element of the vector space $ W $.
- Note that row operations are not needed for this proof; try to use other properties of the vectors or the definition of the vector space $ W $.

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