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 $A$:

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.