# Algebra Examples

Step 1

The kernel of a transformation is a vector that makes the transformation equal to the zero vector (the pre-image of the transformation).

Step 2

Create a system of equations from the vector equation.

Step 3

Write the system as a matrix.

Step 4

Step 4.1

Perform the row operation to make the entry at a .

Step 4.1.1

Perform the row operation to make the entry at a .

Step 4.1.2

Simplify .

Step 4.2

Perform the row operation to make the entry at a .

Step 4.2.1

Perform the row operation to make the entry at a .

Step 4.2.2

Simplify .

Step 4.3

Swap with to put a nonzero entry at .

Step 4.4

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

Step 4.4.1

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

Step 4.4.2

Simplify .

Step 4.5

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

Step 4.5.1

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

Step 4.5.2

Simplify .

Step 4.6

Perform the row operation to make the entry at a .

Step 4.6.1

Perform the row operation to make the entry at a .

Step 4.6.2

Simplify .

Step 4.7

Perform the row operation to make the entry at a .

Step 4.7.1

Perform the row operation to make the entry at a .

Step 4.7.2

Simplify .

Step 4.8

Perform the row operation to make the entry at a .

Step 4.8.1

Perform the row operation to make the entry at a .

Step 4.8.2

Simplify .

Step 5

Use the result matrix to declare the final solution to the system of equations.

Step 6

Write a solution vector by solving in terms of the free variables in each row.

Step 7

Write as a solution set.

Step 8

The kernel of is the subspace .