# Linear Algebra Examples

Step 1

Nullity is the dimension of the null space, which is the same as the number of free variables in the system after row reducing. The free variables are the columns without pivot positions.

Step 2

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

Multiply each element of by to make the entry at a .

Step 2.5.1

Multiply each element of by 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 2.7

Perform the row operation to make the entry at a .

Step 2.7.1

Perform the row operation to make the entry at a .

Step 2.7.2

Simplify .

Step 2.8

Perform the row operation to make the entry at a .

Step 2.8.1

Perform the row operation to make the entry at a .

Step 2.8.2

Simplify .

Step 3

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 4

The nullity is the number of columns without a pivot position in the row reduced matrix.