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Find bases of the row space and the column space of A. What is the rank of A?

To find the bases of the row space and column space of the matrix \( A \), and to determine the rank of \( A \), we can follow these steps:

### 1. Matrix \( A \):
\[
A = \begin{pmatrix}
1 & -1 & 0 & 0 \\
0 & 1 & 1 & 1 \\
1 & -1 & 0 & 0 \\
0 & 2 & 2 & 2 \\
0 & 0 & 0 & 0
\end{pmatrix}
\]

### 2. Row Space
To find the basis of the row space, we need to reduce the matrix \( A \) to its row echelon form (REF).

Let's perform row reduction:

- Subtract row 1 from row 3:
  \[
  R_3 \rightarrow R_3 - R_1 = (1, -1, 0, 0) - (1, -1, 0, 0) = (0, 0, 0, 0)
  \]
  The matrix becomes:
  \[
  \begin{pmatrix}
  1 & -1 & 0 & 0 \\
  0 & 1 & 1 & 1 \\
  0 & 0 & 0 & 0 \\
  0 & 2 & 2 & 2 \\
  0 & 0 & 0 & 0
  \end{pmatrix}
  \]

- Subtract 2 times row 2 from row 4:
  \[
  R_4 \rightarrow R_4 - 2R_2 = (0, 2, 2, 2) - 2(0, 1, 1, 1) = (0, 0, 0, 0)
  \]
  The matrix becomes:
  \[
  \begin{pmatrix}
  1 & -1 & 0 & 0 \\
  0 & 1 & 1 & 1 \\
  0 & 0 & 0 & 0 \\
  0 & 0 & 0 & 0 \\
  0 & 0 & 0 & 0
  \end{pmatrix}
  \]

The row-reduced form shows that the first two rows are linearly independent, and the rest are zero rows. Therefore, the basis for the row space consists of the first two non-zero rows of \( A \):

**Basis for the row space:**
\[
\left\{ \begin{pmatrix} 1 & -1 & 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 1 & 1 & 1 \end{pmatrix} \right\}
\]

### 3. Column Space
To find the basis of the column space, we look at the pivot columns in the row-reduced form of \( A \). The pivot columns correspond to the first and second columns of \( A \). Therefore, the basis for the column space consists of these columns from the original matrix \( A \):

**Basis for the column space:**
\[
\left\{ \begin{pmatrix} 1 \\ 0 \\ 1 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix} -1 \\ 1 \\ -1 \\ 2 \\ 0 \end{pmatrix} \right\}
\]

### 4. Rank of \( A \)
The rank of a matrix is the number of pivot columns in its row echelon form. From the row-reduced form, we see there are 2 pivot columns, so the rank of \( A \) is 2.

**Rank of \( A \):** \( \text{rank}(A) = 2 \)

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Would you like more details on any of these steps or have any questions?

### Related Questions:
1. How do you find the null space of a matrix?
2. What is the relationship between the rank of a matrix and its nullity?
3. How can you determine if a set of vectors is linearly independent?
4. What is the significance of pivot columns in matrix computations?
5. Can a matrix have a different rank if the field is different (e.g., complex vs real numbers)?

**Tip:** Always double-check your row operations when reducing matrices to ensure accuracy in finding the basis and rank.

Discover how to find the bases of the row space and column space of matrix A, along with determining its rank. This problem involves key concepts from linear algebra, including matrix operations and row echelon form. Suitable for advanced undergraduate students studying linear algebra.

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