To find a basis for the column space of the matrix $A$, we need to determine which columns are linearly independent. The matrix $A$ is given as:

$A = \begin{bmatrix} 2 & 4 & -6 & 2 & -4 \\ -1 & -2 & 3 & 1 & 8 \\ 0 & 0 & 0 & 2 & 6 \end{bmatrix}$

Steps to find the basis for the column space:

1. Set up the matrix for row reduction: We will use row operations to bring $A$ to its row echelon form (or reduced row echelon form if needed).
2. Identify the pivot columns: The columns that contain leading ones in the row echelon form are the linearly independent columns and will form the basis for the column space.
3. Extract basis vectors: The original columns of $A$ corresponding to the pivot columns in the row echelon form are the basis vectors.

Solution

Let's go through the row reduction of $A$ (detailed steps are generally worked out on paper or with software for precision, but here’s the process):

Row Echelon Form Calculation

Upon reducing $A$, we get:

$\text{Row Echelon Form} = \begin{bmatrix} 1 & 2 & -3 & 0 & 0 \\ 0 & 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}$

From this form:

• The pivot columns are the first and fourth columns.

Basis for the Column Space

The basis vectors for the column space of $A$ are the columns of $A$ corresponding to the pivot columns (1st and 4th columns):

$\text{Basis} = \begin{bmatrix} 2 & 2 \\ -1 & 1 \\ 0 & 2 \end{bmatrix}$

So, the answer is:

$\begin{bmatrix} 2 & 2 \\ -1 & 1 \\ 0 & 2 \end{bmatrix}$

This matrix represents the basis for the column space of $A$.

Would you like further details on the row reduction process, or do you have any other questions?

Related Questions

1. How do you find the basis for the null space of a matrix?
2. What is the significance of the row echelon form in determining the rank of a matrix?
3. How can you determine if a set of vectors forms a basis for a given space?
4. What are the applications of finding the column space of a matrix in real-world problems?
5. Can a matrix have more than one possible basis for its column space?

Tip

Remember that a basis for the column space of a matrix must include linearly independent vectors only; any dependent columns are omitted from the basis.