To compute the range (or column space) of a matrix, we need to determine the span of its columns. The range of a matrix \( A \) consists of all possible linear combinations of its columns, which geometrically represents the space in which the columns of the matrix lie.

### Steps to Compute the Range:

1. **Write down the matrix \( A \)**.
2. **Perform Gaussian elimination** to bring the matrix to its row-echelon form (REF) or reduced row-echelon form (RREF).
3. **Identify the pivot columns** in the REF or RREF. The pivot columns correspond to the linearly independent columns in the original matrix.
4. **The range of the matrix** is the span of these linearly independent columns in the original matrix.

Let's go through an example.

Suppose you have a matrix \( A \):

\[
A = \begin{pmatrix}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9 \\
\end{pmatrix}
\]

### Step 1: Write down the matrix \( A \).

\[
A = \begin{pmatrix}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9 \\
\end{pmatrix}
\]

### Step 2: Perform Gaussian elimination to get the REF or RREF.

Applying row operations, we obtain:

\[
\text{REF of } A = \begin{pmatrix}
1 & 2 & 3 \\
0 & -3 & -6 \\
0 & 0 & 0 \\
\end{pmatrix}
\]

### Step 3: Identify the pivot columns.

The pivot columns here are the first and second columns (since they contain leading 1's in the REF).

### Step 4: The range of the matrix.

The range is the span of the first and second columns of the original matrix \( A \):

\[
\text{Range}(A) = \text{span} \left( \begin{pmatrix} 1 \\ 4 \\ 7 \end{pmatrix}, \begin{pmatrix} 2 \\ 5 \\ 8 \end{pmatrix} \right)
\]

This means the range of matrix \( A \) is the space formed by all linear combinations of these two vectors.

### Conclusion:
The range of the matrix \( A \) is a subspace of \(\mathbb{R}^3\) spanned by the vectors \( \begin{pmatrix} 1 \\ 4 \\ 7 \end{pmatrix} \) and \( \begin{pmatrix} 2 \\ 5 \\ 8 \end{pmatrix} \).

---

Would you like more details or have any questions?

Here are some related questions:
1. How do you determine if a matrix is full rank?
2. What is the difference between the range and the null space of a matrix?
3. How does the dimension of the range relate to the rank of the matrix?
4. How can the range of a matrix be used in solving systems of linear equations?
5. What is the significance of the rank-nullity theorem in understanding the range?

**Tip:** Always remember that the number of pivot columns (leading 1's) in the row-echelon form of a matrix gives the dimension of the range, also known as the rank of the matrix.

Learn how to compute the range (column space) of a matrix with a step-by-step explanation. This problem covers key concepts in linear algebra, including the span of columns and the application of Gaussian elimination. Suitable for advanced high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation