Examples from Kernel, image and rank All examples from Kernel, image and rank.
[ EXAMPLE ] For
A = ( 2 2 2 0 3 − 3 ) , A=
\begin{pmatrix}
2&2&2\\
0&3&-3
\end{pmatrix}, A = ( 2 0 2 3 2 − 3 ) , let x = ( x 1 , x 2 , x 3 ) T x=(x_1,x_2,x_3)^T x = ( x 1 , x 2 , x 3 ) T A x = 0 Ax=0 A x = 0
2 x 1 + 2 x 2 + 2 x 3 = 0 , 3 x 2 − 3 x 3 = 0. 2x_1+2x_2+2x_3=0,
\qquad
3x_2-3x_3=0. 2 x 1 + 2 x 2 + 2 x 3 = 0 , 3 x 2 − 3 x 3 = 0. Thus x 2 = x 3 x_2=x_3 x 2 = x 3 x 1 = − 2 x 3 x_1=-2x_3 x 1 = − 2 x 3 x 3 = λ x_3=\lambda x 3 = λ
ker ( A ) = { λ ( − 2 1 1 ) : λ ∈ R } = span { ( − 2 1 1 ) } . \ker(A)
=
\left\{
\lambda
\begin{pmatrix}
-2\\1\\1
\end{pmatrix}
:\lambda\in\mathbb R
\right\}
=
\operatorname{span}
\left\{
\begin{pmatrix}
-2\\1\\1
\end{pmatrix}
\right\}. ker ( A ) = ⎩ ⎨ ⎧ λ − 2 1 1 : λ ∈ R ⎭ ⎬ ⎫ = span ⎩ ⎨ ⎧ − 2 1 1 ⎭ ⎬ ⎫ . [RIGHT CLICK] [ASK AI] [ EXAMPLE ] For the matrix
A = ( 2 2 2 0 3 − 3 ) , A=
\begin{pmatrix}
2&2&2\\
0&3&-3
\end{pmatrix}, A = ( 2 0 2 3 2 − 3 ) , the image is the span of its columns. The first two columns,
( 2 0 ) , ( 2 3 ) , \begin{pmatrix}2\\0\end{pmatrix},
\qquad
\begin{pmatrix}2\\3\end{pmatrix}, ( 2 0 ) , ( 2 3 ) , are linearly independent: if c 1 ( 2 , 0 ) T + c 2 ( 2 , 3 ) T = 0 c_1(2,0)^T+c_2(2,3)^T=0 c 1 ( 2 , 0 ) T + c 2 ( 2 , 3 ) T = 0 c 2 = 0 c_2=0 c 2 = 0 c 1 = 0 c_1=0 c 1 = 0 R 2 \mathbb R^2 R 2
im ( A ) = R 2 . \operatorname{im}(A)=\mathbb R^2. im ( A ) = R 2 . [RIGHT CLICK] [ASK AI] Kernel, image and rank Builds on Subspaces and 3 more.