Question

I need to understand how to solve questions like this: Find a basis for the range and a basis for the nullspace of the following linear mapping: \(\sf L(x_1,x_2,x_3)=(2x_1, -x_2+2x_3)\)

I understand a little bit of nullspace, but i'm still not sure about the range of linear mapping.

Answer 1

@Nnesha @jim_thompson5910 @jigglypuff314

Answer 2

@zepdrix

Answer 3

\[\begin{pmatrix}2&0&0\\0&-1&2\end{pmatrix}\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}=\begin{pmatrix}2x_1\\-x_2+2x_3\end{pmatrix}\]

Answer 4

given Ax=b

the columns of A, span the range ... and a basis consists of all the independant column vectors

Answer 5

the basis for the null set, is determined by the solutions to:
\[\begin{pmatrix}2&0&0\\0&-1&2\end{pmatrix}\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}=\begin{pmatrix}2x_1\color{red}{=0}\\-x_2+2x_3\color{red}{=0}\end{pmatrix}\]