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We are tasked with finding the bases for the **row space**, **column space**, and **null space** of matrix \( A \), where:

\[
A = \begin{bmatrix}
1 & -4 & 9 & -7 \\
-1 & 2 & -4 & 1 \\
5 & 6 & 10 & 7
\end{bmatrix}
\]

### Steps:

1. **Row Space**: To find the basis of the row space, we perform **row reduction** on \( A \) to bring it into **row echelon form (REF)** or **reduced row echelon form (RREF)**. The non-zero rows of the resulting matrix form the basis of the row space.

2. **Column Space**: To find the basis of the column space, we can either:
   - Identify the pivot columns in the row-reduced matrix.
   - Alternatively, we can perform column reduction or inspect the original matrix to determine linearly independent columns.

3. **Null Space**: The null space consists of the solutions to the equation \( A \mathbf{x} = 0 \). We will solve this system of equations using Gaussian elimination to find the general solution, which will give us the basis for the null space.

Let me compute these steps for you.### Results:

1. **Row Space Basis** (from the row-reduced form):
   The matrix in row-reduced form (RREF) is:
   \[
   \begin{bmatrix}
   1 & 0 & 0 & 0 \\
   0 & 1 & 0 & 0 \\
   0 & 0 & 1 & 0
   \end{bmatrix}
   \]
   Therefore, the **basis for the row space** is the first three rows of the original matrix \( A \):
   \[
   \begin{bmatrix}
   1 & -4 & 9 & -7 \\
   -1 & 2 & -4 & 1 \\
   5 & 6 & 10 & 7
   \end{bmatrix}
   \]

2. **Column Space Basis**:
   The pivot columns from the RREF indicate that the first, second, and third columns are linearly independent. Hence, the **basis for the column space** is:
   \[
   \begin{bmatrix}
   1 & -4 & 9 \\
   -1 & 2 & -4 \\
   5 & 6 & 10
   \end{bmatrix}
   \]

3. **Null Space Basis**:
   The **basis for the null space** is given by the vector:
   \[
   \begin{bmatrix}
   -3.8 \\
   0 \\
   1.2 \\
   1
   \end{bmatrix}
   \]
   This represents the general solution to \( A \mathbf{x} = 0 \).

Would you like further clarification on any of the steps?

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Here are 5 related questions to deepen your understanding:
1. What is the geometric interpretation of the row space and column space?
2. How do the dimensions of the row space and column space relate to the rank of the matrix?
3. How can you verify if a given vector is in the null space of a matrix?
4. How does the rank-nullity theorem connect the null space and row space?
5. What are the implications of the matrix being invertible in terms of its null space?

**Tip**: The row space and column space always have the same dimension, which is the rank of the matrix.

Find the bases for the row space, the column space & Null space of A, where A is the matrix: [1, -4, 9, -7; -1, 2, -4, 1; 5, 6, 10, 7]

This problem walks through how to find the bases for the row space, column space, and null space of a given matrix A using Gaussian elimination. It includes key linear algebra concepts such as row reduction and the rank-nullity theorem, suitable for undergraduate students.

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