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(d) Find a basis for the column space of *A*.

To find a **basis for the column space** of matrix \(A\), we first need to determine which columns are linearly independent. Since the matrices \(A\) and \(B\) are given to be **row-equivalent**, both have the same rank, which is 3. This means that exactly 3 columns of \(A\) are linearly independent, and these columns will form a basis for the **column space** of \(A\). 

Here’s the step-by-step approach to determine the basis:

### 1. Identify Pivot Columns from \(B\)
Matrix \(B\) is in **row echelon form**:
\[
B = 
\begin{pmatrix}
1 & 0 & 0 & 1 & 1 \\
0 & 1 & -2 & 0 & 3 \\
0 & 0 & 1 & 0 & -5 \\
0 & 0 & 0 & 0 & 0
\end{pmatrix}
\]
From the non-zero rows of \(B\), the **pivot columns** (the ones with the leading 1 in each row) are the **1st, 2nd, and 3rd columns**. These columns correspond to the independent columns in matrix \(A\).

### 2. Extract the Basis Columns from \(A\)
Now, we take the **1st, 2nd, and 3rd columns** from matrix \(A\):
\[
A = 
\begin{pmatrix}
-2 & -5 & 8 & 0 & -17 \\
1 & 3 & -5 & 1 & 5 \\
1 & 5 & -9 & 5 & -9 \\
1 & 7 & -13 & 5 & -3
\end{pmatrix}
\]
So, the first three columns of \(A\) are:
\[
\text{Column 1:} \quad \begin{pmatrix} -2 \\ 1 \\ 1 \\ 1 \end{pmatrix}, \quad
\text{Column 2:} \quad \begin{pmatrix} -5 \\ 3 \\ 5 \\ 7 \end{pmatrix}, \quad
\text{Column 3:} \quad \begin{pmatrix} 8 \\ -5 \\ -9 \\ -13 \end{pmatrix}
\]

### 3. Write the Basis for the Column Space
The set of these three columns forms a basis for the **column space of \(A\)**:
\[
\left\{ 
\begin{pmatrix} -2 \\ 1 \\ 1 \\ 1 \end{pmatrix}, 
\begin{pmatrix} -5 \\ 3 \\ 5 \\ 7 \end{pmatrix}, 
\begin{pmatrix} 8 \\ -5 \\ -9 \\ -13 \end{pmatrix} 
\right\}
\]

### Final Answer
The **basis for the column space of \(A\)** is:
\[
\left\{ 
\begin{pmatrix} -2 \\ 1 \\ 1 \\ 1 \end{pmatrix}, 
\begin{pmatrix} -5 \\ 3 \\ 5 \\ 7 \end{pmatrix}, 
\begin{pmatrix} 8 \\ -5 \\ -9 \\ -13 \end{pmatrix} 
\right\}
\]

This is a set of three linearly independent vectors in \(\mathbb{R}^4\), which spans the column space of \(A\).

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Do you have any questions or need further clarifications?

Here are 5 related questions:
1. How can you determine the rank of a matrix without row reduction?
2. What is the dimension of the null space of \(A\) given its nullity?
3. How do row equivalence and column equivalence differ?
4. How would you find a basis for the **null space** of \(A\)?
5. Can two matrices with the same rank have different column spaces?

**Tip:** A basis is not unique; different sets of independent columns can form valid bases for the same column space.

Learn how to find a basis for the column space of matrix A by identifying linearly independent columns through row equivalence. Understand key linear algebra concepts such as rank, nullity, and basis formation. This solution includes a step-by-step approach to identifying pivot columns and constructing a basis for the column space of A.

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