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.
Tap for more steps...
Step 4.1
Perform the row operation to make the entry at a .
Tap for more steps...
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 .
Tap for more steps...
Step 4.2.1
Perform the row operation to make the entry at a .
Step 4.2.2
Simplify .
Step 4.3
Multiply each element of by to make the entry at a .
Tap for more steps...
Step 4.3.1
Multiply each element of by to make the entry at a .
Step 4.3.2
Simplify .
Step 4.4
Perform the row operation to make the entry at a .
Tap for more steps...
Step 4.4.1
Perform the row operation to make the entry at a .
Step 4.4.2
Simplify .
Step 4.5
Perform the row operation to make the entry at a .
Tap for more steps...
Step 4.5.1
Perform the row operation to make the entry at a .
Step 4.5.2
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 the solution as a linear combination of vectors.
Step 8
Write as a solution set.
Step 9
The solution is the set of vectors created from the free variables of the system.
Step 10
The kernel of is the subspace .
Enter YOUR Problem
Mathway requires javascript and a modern browser.