how can the basis of a column space be written as a linear independent sets of A

To write the basis of a column space of a matrix \( A \) as a linearly independent set, follow these steps:

### 1. Identify the Column Space
The **column space** of a matrix \( A \) is the span of its columns. It consists of all possible linear combinations of the columns of \( A \).

### 2. Perform Gaussian Elimination (or Row Reduction)
- Start by performing Gaussian elimination (row reduction) on matrix \( A \) to obtain its row echelon form (REF).
- The key is to track the original columns of \( A \) that correspond to the pivot columns in the REF. Pivot columns are the ones that contain the leading entries (the first non-zero entries in each row) after performing row reduction.

### 3. Identify Pivot Columns
- Identify the columns in the original matrix \( A \) that correspond to the pivot columns in the REF.
- The columns of \( A \) that correspond to these pivot positions form a **linearly independent set**. 

### 4. Form the Basis
- The columns of \( A \) corresponding to the pivot positions are a **basis** for the column space of \( A \). This means that these columns are linearly independent and span the column space of \( A \).

### Example

Consider the matrix \( A \):

\[
A = \begin{pmatrix}
1 & 2 & 3 \\
2 & 4 & 6 \\
4 & 8 & 12
\end{pmatrix}
\]

#### Step 1: Row Reduce \( A \)

Row reducing \( A \) to REF:

\[
\text{REF}(A) = \begin{pmatrix}
1 & 2 & 3 \\
0 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix}
\]

#### Step 2: Identify Pivot Columns
- The pivot column is the first column in this case.

#### Step 3: Form the Basis
- The basis for the column space of \( A \) is the set containing the first column of \( A \):

\[
\text{Basis} = \left\{ \begin{pmatrix} 1 \\ 2 \\ 4 \end{pmatrix} \right\}
\]

This vector is linearly independent and spans the column space of \( A \).

Would you like more details or have any questions?

### Related Questions:

1. What is the dimension of the column space, and how does it relate to the rank of the matrix?
2. How do you determine if a set of vectors is linearly independent?
3. What is the difference between the column space and the null space of a matrix?
4. How do you find the row space of a matrix, and how is it related to the column space?
5. Can a matrix have more than one basis for its column space?
6. What is the significance of the column space in solving linear systems?
7. How do you determine if a vector is in the column space of a matrix?
8. What is the connection between the column space and the determinant of a matrix?

### Tip:
Always keep track of the original columns when performing row reduction to ensure you correctly identify the basis of the column space.

Learn how to write the basis of a column space of matrix A as a linearly independent set using Gaussian elimination and row reduction. This detailed explanation includes an example matrix and step-by-step instructions.

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