# Algebra Examples

Find the Basis and Dimension for the Column Space of the Matrix
Step 1
Find the reduced row echelon form.
Step 1.1
Multiply each element of by to make the entry at a .
Step 1.1.1
Multiply each element of by to make the entry at a .
Step 1.1.2
Simplify .
Step 1.2
Perform the row operation to make the entry at a .
Step 1.2.1
Perform the row operation to make the entry at a .
Step 1.2.2
Simplify .
Step 1.3
Multiply each element of by to make the entry at a .
Step 1.3.1
Multiply each element of by to make the entry at a .
Step 1.3.2
Simplify .
Step 1.4
Perform the row operation to make the entry at a .
Step 1.4.1
Perform the row operation to make the entry at a .
Step 1.4.2
Simplify .
Step 1.5
Perform the row operation to make the entry at a .
Step 1.5.1
Perform the row operation to make the entry at a .
Step 1.5.2
Simplify .
Step 2
The pivot positions are the locations with the leading in each row. The pivot columns are the columns that have a pivot position.
Pivot Positions: and
Pivot Columns: and
Step 3
The basis for the column space of a matrix is formed by considering corresponding pivot columns in the original matrix. The dimension of is the number of vectors in a basis for .
Basis of :
Dimension of :