Linear Algebra Examples

$[\begin{array}{c} 2 & 3 & 4 \\ 1 & 2 & 1 \\ 3 & 1 & 2 \end{array}] = [\begin{array}{c} x \\ y \\ z \end{array}]$

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).

$[\begin{array}{c} x \\ y \\ z \end{array}] = 0$

Step 2

Create a system of equations from the vector equation.

$x = 0$

$y = 0$

$z = 0$

Step 3

Write the system as a matrix.

$[\begin{array}{c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}]$

Step 4

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

$x = 0$

$y = 0$

$z = 0$

Step 5

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

$[\begin{array}{c} x \\ y \\ z \end{array}] = [\begin{array}{c} 0 \\ 0 \\ 0 \end{array}]$

Step 6

Write as a solution set.

${[\begin{array}{c} 0 \\ 0 \\ 0 \end{array}]}$

Step 7

The kernel of $M$ is the subspace ${[\begin{array}{c} 0 \\ 0 \\ 0 \end{array}]}$ .

$K (M) = {[\begin{array}{c} 0 \\ 0 \\ 0 \end{array}]}$