Linear Algebra Examples

[\begin{array}{c} a & b \\ c & d \end{array}] \cdot [\begin{array}{c} 4 \\ 1 \end{array}] = [\begin{array}{c} 0 \\ 0 \end{array}]

$[\begin{array}{c} a & b \\ c & d \end{array}] \cdot [\begin{array}{c} 4 \\ 1 \end{array}] = [\begin{array}{c} 0 \\ 0 \end{array}]$

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} 0 \\ 0 \end{array}] = 0

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

Step 2

Create a system of equations from the vector equation.

0 = 0

$0 = 0$

0 = 0

$0 = 0$

Step 3

Write the system of equations in matrix form.

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

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

Step 4

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

Step 5

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

{}

${}$

Step 6

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.

X = = [\begin{array}{c} 0 \end{array}]

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

Step 7

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

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

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

Step 8

The kernel of

M

$M$ is the subspace

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

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

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

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