Hey guys!

I have a question here where I'm supposed to get the kernel and image of a linear transformation. I think I must be missing something, coz I feel like I haven't done enough!

Let T: $\displaystyle R^3$$\displaystyle \rightarrow$$\displaystyle R^2$ be the linear transformation defined by $\displaystyle T((x_{1},x_{2},x_{3}))=(x_{1}+x_{2},x_{2}-2x_{3})$

(a) Find the kernel of T

(b) Show that (1,0) and (0,1) are in the range of T. Determine the range of T.

For (a), I just let:

$\displaystyle x_{1}+x_{2}=0$,and

$\displaystyle x_{2}-2x_{3}=0$,

and I get $\displaystyle x_{1}=-2x_{3}$.

So is my answer: kerT= $\displaystyle (-2x_{3}+x_{2},x_{2}+x_{1})$

I'm guessing this is a stupid question, and I'm missing out loads! Help would be appreciated!

Thanks,

SimplySparklers