# Finite Math 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.
Step 8
Check if the vectors are linearly independent.
Step 8.1
List the vectors.
Step 8.2
Write the vectors as a matrix.
Step 8.3
To determine if the columns in the matrix are linearly dependent, determine if the equation has any non-trivial solutions.
Step 8.4
Write as an augmented matrix for .
Step 8.5
Find the reduced row echelon form.
Step 8.5.1
Multiply each element of by to make the entry at a .
Step 8.5.1.1
Multiply each element of by to make the entry at a .
Step 8.5.1.2
Simplify .
Step 8.5.2
Perform the row operation to make the entry at a .
Step 8.5.2.1
Perform the row operation to make the entry at a .
Step 8.5.2.2
Simplify .
Step 8.5.3
Multiply each element of by to make the entry at a .
Step 8.5.3.1
Multiply each element of by to make the entry at a .
Step 8.5.3.2
Simplify .
Step 8.5.4
Perform the row operation to make the entry at a .
Step 8.5.4.1
Perform the row operation to make the entry at a .
Step 8.5.4.2
Simplify .
Step 8.5.5
Perform the row operation to make the entry at a .
Step 8.5.5.1
Perform the row operation to make the entry at a .
Step 8.5.5.2
Simplify .
Step 8.5.6
Multiply each element of by to make the entry at a .
Step 8.5.6.1
Multiply each element of by to make the entry at a .
Step 8.5.6.2
Simplify .
Step 8.5.7
Perform the row operation to make the entry at a .
Step 8.5.7.1
Perform the row operation to make the entry at a .
Step 8.5.7.2
Simplify .
Step 8.5.8
Perform the row operation to make the entry at a .
Step 8.5.8.1
Perform the row operation to make the entry at a .
Step 8.5.8.2
Simplify .
Step 8.5.9
Perform the row operation to make the entry at a .
Step 8.5.9.1
Perform the row operation to make the entry at a .
Step 8.5.9.2
Simplify .
Step 8.5.10
Perform the row operation to make the entry at a .
Step 8.5.10.1
Perform the row operation to make the entry at a .
Step 8.5.10.2
Simplify .
Step 8.5.11
Perform the row operation to make the entry at a .
Step 8.5.11.1
Perform the row operation to make the entry at a .
Step 8.5.11.2
Simplify .
Step 8.6
Remove rows that are all zeros.
Step 8.7
Write the matrix as a system of linear equations.
Step 8.8
Since the only solution to is the trivial solution, the vectors are linearly independent.
Linearly Independent
Linearly Independent
Step 9
Since the vectors are linearly independent, they form a basis for the null space of the matrix.
Basis of :
Dimension of :