Find the basis of a column space and write linear combination

To find the basis of a column space of a matrix and express each column as a linear combination of the basis vectors, follow these steps:

### Step 1: Identify the Matrix
Let’s denote the matrix as \( A \). For illustration, assume:
\[
A = \begin{pmatrix}
1 & 2 & 3 \\
2 & 4 & 6 \\
3 & 6 & 9
\end{pmatrix}
\]

### Step 2: Perform Gaussian Elimination
Convert the matrix to its row echelon form (REF) or reduced row echelon form (RREF) by performing row operations.

**RREF of \( A \):**
\[
R = \begin{pmatrix}
1 & 2 & 3 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix}
\]

### Step 3: Identify the Pivot Columns
The columns corresponding to the pivot positions (non-zero rows in the REF) form the basis for the column space. In the matrix \( R \), the pivot is in the first column.

### Step 4: Extract the Basis Vectors
The basis for the column space of \( A \) is the set of original columns in \( A \) corresponding to the pivot columns in \( R \). Here, the first column of \( A \) is the only pivot column.

Thus, the basis for the column space of \( A \) is:
\[
\text{Basis} = \left\{ \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} \right\}
\]

### Step 5: Write Linear Combinations
Each column in the original matrix \( A \) can be expressed as a linear combination of the basis vectors:

- First column: \( \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} = 1 \times \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} \)
- Second column: \( \begin{pmatrix} 2 \\ 4 \\ 6 \end{pmatrix} = 2 \times \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} \)
- Third column: \( \begin{pmatrix} 3 \\ 6 \\ 9 \end{pmatrix} = 3 \times \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} \)

So, the basis is \( \left\{ \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} \right\} \), and each column is a scalar multiple of this basis vector.

---

Would you like more details or have any questions?

Here are some related questions to consider:
1. What is the dimension of the column space?
2. How do you determine if a set of vectors is linearly independent?
3. Can a matrix have more than one basis for its column space?
4. What is the relationship between the rank of a matrix and its column space?
5. How do you find the null space of a matrix?
6. How do row operations affect the column space?
7. What is the geometric interpretation of the column space?
8. How can you find a basis for the row space of a matrix?

**Tip:** Always ensure the set of basis vectors is linearly independent to correctly represent the column space.

Discover how to find the basis of a column space and express each column as a linear combination of basis vectors. This detailed explanation includes step-by-step instructions using a matrix example. Ideal for advanced undergraduate students studying linear algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation