Linear Algebra

**smiler** · April 26th 2008, 03:09 AM

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.

**flyingsquirrel** · April 26th 2008, 03:23 AM

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 $\lambda \in \mathbb{R}, \,x=\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}\in\m athbb{R}^3$ , $f(\lambda x)=\lambda f(x)$

**Isomorphism** · April 26th 2008, 08:03 AM

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

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