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
Find the reduced row echelon form.
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 .