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

Create a system of equations from the vector equation.

Write the system of equations in matrix form.

Perform the row operation on (row ) in order to convert some elements in the row to .

Replace (row ) with the row operation in order to convert some elements in the row to the desired value .

Replace (row ) with the actual values of the elements for the row operation .

Simplify (row ).

Perform the row operation on (row ) in order to convert some elements in the row to .

Replace (row ) with the row operation in order to convert some elements in the row to the desired value .

Replace (row ) with the actual values of the elements for the row operation .

Simplify (row ).

Perform the row operation on (row ) in order to convert some elements in the row to .

Replace (row ) with the row operation in order to convert some elements in the row to the desired value .

Replace (row ) with the actual values of the elements for the row operation .

Simplify (row ).

Perform the row operation on (row ) in order to convert some elements in the row to .

Replace (row ) with the actual values of the elements for the row operation .

Simplify (row ).

Perform the row operation on (row ) in order to convert some elements in the row to .

Replace (row ) with the actual values of the elements for the row operation .

Simplify (row ).

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

This expression is the solution set for the system of equations.

Decompose a solution vector by re-arranging each equation represented in the row-reduced form of the augmented matrix by solving for the dependent variable in each row yields the vector equality.

Express the vector as a linear combination of column vector using the properties of vector column addition.

The null space of the set is the set of vectors created from the free variables of the system.

The kernel of is the subspace .

Find the Kernel S([[a],[b],[c]])=[[a-2b-c],[3a-b+2c],[a+b+2c]]