Let AequalsStart 3 By 3 Table 1st Row 1st Column 1 2nd Column 0 3rd Column negative 6 2nd Row 1st Column 0 2nd Column 2 3rd Column negative 2 3rd Row 1st Column negative 3 2nd Column 8 3rd Column 4 EndTable 
  and Bold bequalsStart 3 By 1 Table 1st Row 1st Column 8 2nd Row 1st Column negative 2 3rd Row 1st Column negative 20 EndTable 
 . Denote the columns of A by Bold a 1​, Bold a 2​, Bold a 3​, and let WequalsSpan StartSet Bold a 1 comma Bold a 2 comma Bold a 3 EndSet.
a. Is Bold b in StartSet Bold a 1 comma Bold a 2 comma Bold a 3 EndSet​? How many vectors are in StartSet Bold a 1 comma Bold a 2 comma Bold a 3 EndSet​?
b. Is Bold b in​ W? How many vectors are in​ W?
c. Show that Bold a 2 is in W. ​[​Hint: Row operations are​ unnecessary.]

Given the matrix \( A \) and vector \( \mathbf{b} \), let's break down the problem step-by-step.

### Matrix \( A \) and vector \( \mathbf{b} \):

The matrix \( A \) is:

\[
A = \begin{bmatrix}
1 & 0 & -6 \\
0 & 2 & -2 \\
-3 & 8 & 4
\end{bmatrix}
\]

The vector \( \mathbf{b} \) is:

\[
\mathbf{b} = \begin{bmatrix} 8 \\ -2 \\ -20 \end{bmatrix}
\]

### Part (a): Is \( \mathbf{b} \) in \( \{\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3\} \)? How many vectors are in \( \{\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3\} \)?

The vectors \( \mathbf{a}_1 \), \( \mathbf{a}_2 \), and \( \mathbf{a}_3 \) are the column vectors of \( A \):

\[
\mathbf{a}_1 = \begin{bmatrix} 1 \\ 0 \\ -3 \end{bmatrix}, \quad
\mathbf{a}_2 = \begin{bmatrix} 0 \\ 2 \\ 8 \end{bmatrix}, \quad
\mathbf{a}_3 = \begin{bmatrix} -6 \\ -2 \\ 4 \end{bmatrix}
\]

Now, the question is asking whether \( \mathbf{b} \) is one of these vectors. To check this, compare \( \mathbf{b} \) with \( \mathbf{a}_1 \), \( \mathbf{a}_2 \), and \( \mathbf{a}_3 \):

- \( \mathbf{b} \neq \mathbf{a}_1 \)
- \( \mathbf{b} \neq \mathbf{a}_2 \)
- \( \mathbf{b} \neq \mathbf{a}_3 \)

Thus, \( \mathbf{b} \) is not in \( \{\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3\} \).

**Number of vectors** in

Let A equal Start 3 By 3 Table 1st Row 1st Column 1 2nd Column 0 3rd Column negative 6 2nd Row 1st Column 0 2nd Column 2 3rd Column negative 2 3rd Row 1st Column negative 3 2nd Column 8 3rd Column 4 End Table and Bold b equal Start 3 By 1 Table 1st Row 1st Column 8 2nd Row 1st Column negative 2 3rd Row 1st Column negative 20 End Table. Denote the columns of A by Bold a 1, Bold a 2, Bold a 3, and let W equal Span Start Set Bold a 1 comma Bold a 2 comma Bold a 3 End Set. a. Is Bold b in Start Set Bold a 1 comma Bold a 2 comma Bold a 3 End Set? How many vectors are in Start Set Bold a 1 comma Bold a 2 comma Bold a 3 End Set? b. Is Bold b in W? How many vectors are in W? c. Show that Bold a 2 is in W. ​[Hint: Row operations are​ unnecessary.]

Explore a detailed solution to a Linear Algebra problem involving matrix operations and vector spans. Learn how to check if a vector is part of a span and how to demonstrate that a column is in the span of other vectors. This solution is ideal for undergraduate students studying vector spaces and matrix theory.

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Solution

Matrix $A$ and vector $\mathbf{b}$ :

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