To find the basis for the column space of a matrix $A$, you need to follow these steps:

1. Identify the columns of $A$: Begin with the matrix $A$ and note all of its columns.

2. Reduce $A$ to its row echelon form (REF) or reduced row echelon form (RREF): This process will help determine which columns are linearly independent, as pivot columns indicate the ones that contribute uniquely to the span of the column space.

3. Identify the pivot columns: In the REF or RREF of $A$, locate the columns with leading entries (the "pivots"). These correspond to the columns in the original matrix $A$ that form the basis for the column space.

4. Select the original columns: The basis for the column space of $A$ consists of the original columns of $A$ that correspond to the pivot columns in its REF or RREF.

5. Write the basis: The resulting set of vectors (columns) forms the basis for $\text{Col}(A)$, the column space of $A$.

Example

Suppose you have the matrix: $A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}$

1. Perform row reduction on $A$ to identify pivot columns.
2. Let's assume after row reduction, columns 1 and 2 are pivot columns.
3. The basis for $\text{Col}(A)$ is the set of vectors in the original matrix corresponding to these columns: $\{ \begin{bmatrix} 1 \\ 4 \\ 7 \end{bmatrix}, \begin{bmatrix} 2 \\ 5 \\ 8 \end{bmatrix} \}$.

Would you like a detailed example with calculations, or have any specific matrix in mind?

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Related Questions

1. How do I find a basis for the null space of a matrix?
2. What is the dimension of a column space?
3. How do I determine if a set of vectors spans a space?
4. How is the rank of a matrix related to its column space?
5. Can a row reduce be used to determine independence in rows as well as columns?

Tip: Always remember that the basis of a space is a minimal spanning set of linearly independent vectors.