# Algebra Examples

Find the Basis and Dimension for the Null Space of the Matrix
Step 1
Write as an augmented matrix for .
Step 2
Find the reduced row echelon form.
Step 2.1
Multiply each element of by to make the entry at a .
Step 2.1.1
Multiply each element of by to make the entry at a .
Step 2.1.2
Simplify .
Step 2.2
Perform the row operation to make the entry at a .
Step 2.2.1
Perform the row operation to make the entry at a .
Step 2.2.2
Simplify .
Step 2.3
Perform the row operation to make the entry at a .
Step 2.3.1
Perform the row operation to make the entry at a .
Step 2.3.2
Simplify .
Step 2.4
Multiply each element of by to make the entry at a .
Step 2.4.1
Multiply each element of by to make the entry at a .
Step 2.4.2
Simplify .
Step 2.5
Perform the row operation to make the entry at a .
Step 2.5.1
Perform the row operation to make the entry at a .
Step 2.5.2
Simplify .
Step 2.6
Perform the row operation to make the entry at a .
Step 2.6.1
Perform the row operation to make the entry at a .
Step 2.6.2
Simplify .
Step 3
Use the result matrix to declare the final solution to the system of equations.
Step 4
Write a solution vector by solving in terms of the free variables in each row.
Step 5
Write the solution as a linear combination of vectors.
Step 6
Write as a solution set.
Step 7
The solution is the set of vectors created from the free variables of the system.
Basis of :
Dimension of :