20 6 10 Let A = -1 8 5 and let b = B- 1 -2 1 3 W be the set of all linear combinations of the columns of A. a. Is b in W? b. Show that the third column of A is in W? 3 and

Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter4: Vector Spaces
Section4.5: Basis And Dimension
Problem 66E
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### Linear Algebra - Vector Spaces

Given the matrix \( A \) and vector \( \mathbf{b} \):

\[ A = 
\begin{bmatrix}
2 & 0 & 6 \\
-1 & 8 & 5 \\
1 & -2 & 1 
\end{bmatrix}
\]

\[ \mathbf{b} = 
\begin{bmatrix}
10 \\
3 \\
3 
\end{bmatrix}
\]

Let \( W \) be the set of all linear combinations of the columns of \( A \).

#### Questions to Consider:
**a. Is \(\mathbf{b}\) in \(W\)?**

**b. Show that the third column of \( A \) is in \( W \)?**

#### Explanation:

1. **Vector \( \mathbf{b} \) in \( W \):**

   To determine if \(\mathbf{b}\) is in \( W \), we need to solve the equation \( A\mathbf{x} = \mathbf{b} \) for some vector \(\mathbf{x}\). If a solution \(\mathbf{x}\) exists, then \(\mathbf{b}\) is a linear combination of the columns of \( A \), and thus is in \( W \).

2. **Third Column of \( A \) in \( W \):**

   The third column of \( A \) can be written as:
   
   \[
   \mathbf{a}_3 = 
   \begin{bmatrix} 
   6 \\ 
   5 \\ 
   1 
   \end{bmatrix}
   \]
   
   To show that \(\mathbf{a}_3\) is in \( W \), we need to express \(\mathbf{a}_3\) as a linear combination of the columns of \( A \). This means finding scalars \( c_1 \), \( c_2 \), and \( c_3 \) such that:
   
   \[
   c_1 \mathbf{a}_1 + c_2 \mathbf{a}_2 + c_3 \mathbf{a}_3 = \mathbf{a}_3
   \]
   
   where \( \mathbf{a}_1 \), \( \mathbf{a}_2 \), and \( \mathbf{a}_3 \) are the first, second, and third columns of
Transcribed Image Text:### Linear Algebra - Vector Spaces Given the matrix \( A \) and vector \( \mathbf{b} \): \[ A = \begin{bmatrix} 2 & 0 & 6 \\ -1 & 8 & 5 \\ 1 & -2 & 1 \end{bmatrix} \] \[ \mathbf{b} = \begin{bmatrix} 10 \\ 3 \\ 3 \end{bmatrix} \] Let \( W \) be the set of all linear combinations of the columns of \( A \). #### Questions to Consider: **a. Is \(\mathbf{b}\) in \(W\)?** **b. Show that the third column of \( A \) is in \( W \)?** #### Explanation: 1. **Vector \( \mathbf{b} \) in \( W \):** To determine if \(\mathbf{b}\) is in \( W \), we need to solve the equation \( A\mathbf{x} = \mathbf{b} \) for some vector \(\mathbf{x}\). If a solution \(\mathbf{x}\) exists, then \(\mathbf{b}\) is a linear combination of the columns of \( A \), and thus is in \( W \). 2. **Third Column of \( A \) in \( W \):** The third column of \( A \) can be written as: \[ \mathbf{a}_3 = \begin{bmatrix} 6 \\ 5 \\ 1 \end{bmatrix} \] To show that \(\mathbf{a}_3\) is in \( W \), we need to express \(\mathbf{a}_3\) as a linear combination of the columns of \( A \). This means finding scalars \( c_1 \), \( c_2 \), and \( c_3 \) such that: \[ c_1 \mathbf{a}_1 + c_2 \mathbf{a}_2 + c_3 \mathbf{a}_3 = \mathbf{a}_3 \] where \( \mathbf{a}_1 \), \( \mathbf{a}_2 \), and \( \mathbf{a}_3 \) are the first, second, and third columns of
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