1. ## Linear Algebra

Is anyone able to help me do the following proof please:

Let f:R3 > R3 be given by

f(x1,x2,x3)=(x3,x1+x2,x1-x2)

Prove that f is a linear transformation.

Many thanks to anyone who can help.

2. Hi

You have to prove two things :
• For all $x=\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix},\,y=\b egin{pmatrix}y_1\\y_2\\y_3\end{pmatrix} \in \mathbb{R}^3$, $f(x+y)=f(x)+f(y)$
• For all $\lambda \in \mathbb{R}, \,x=\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}\in\m athbb{R}^3$, $f(\lambda x)=\lambda f(x)$

3. Originally Posted by flyingsquirrel
Hi

You have to prove two things :
• For all $x=\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix},\,y=\b egin{pmatrix}y_1\\y_2\\y_3\end{pmatrix} \in \mathbb{R}^3$, $f(x+y)=f(x)+f(y)$
• For all $\lambda \in \mathbb{R}, \,x=\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}\in\m athbb{R}^3$, $f(\lambda x)=\lambda f(x)$
Or obtain a matrix A such that f(x) = Ax(You should know the result that this is always a linear map).

If $x=\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}\in\mat hbb{R}^3, \,\, A = \begin{pmatrix}0&0&1\\1&1&0\\1&-1&0\end{pmatrix}$