Álgebra lineal Ejemplos

T [\begin{array}{c} x \\ y \end{array}] = [\begin{array}{c} x \\ y \end{array}]

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

Paso 1

El núcleo de una transformación es un vector que hace que la transformación sea igual al vector nulo (la imagen previa de la transformación).

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

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

Paso 2

Crea un sistema de ecuaciones a partir de la ecuación vectorial.

x = 0

$x = 0$

y = 0

$y = 0$

Paso 3

Write the system as a matrix.

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

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

Paso 4

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

x = 0

$x = 0$

y = 0

$y = 0$

Paso 5

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

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

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

Paso 6

Write as a solution set.

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

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

Paso 7

El núcleo de

T

$T$ es el subespacio

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

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

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

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