Linear maps (complex)

Question

enter image description here $(1)$ Let $a, b\in \mathbb C$ and $\alpha: \mathbb C^2 \to \mathbb C$ be given by $(x, y)\mapsto ax + by.$

$\quad(a)\quad$ Show that $\alpha$ is a $\mathbb C$- linear map. What condition(s) you have to check?

$\quad(b)\quad$ For what values of $a$ and $b$ is $\alpha$ surjective? Justify your answer.

$\quad (c)\quad$ Find a basis for $\operatorname{ker}\alpha$. Justify your answer. The answer will depend on $a$ and $b$.

Can anyone help me? I think I got the 1. a) but don't really get linear maps.

So we can help you with respect to understanding linear maps, it would help if you include what you "got" for $1.a$, so we can confirm, or correct your work, and go from there. — amWhy, Nov 30 '13 at 16:56
Yeah, but don't know what to do next. or is this right at least? — mrHarry, Nov 30 '13 at 21:36

score 2 · Answer 1 · answered Nov 30 '13 at 17:22

Hint:

For any linear space $\;V_{\Bbb F}\;$ over a field, we have that a linear functional $\;f:V\to\Bbb F\;$ is either the zero function or automatically surjective, because of the dimensions theorem:

$$\dim_{\Bbb F} V=\dim\ker f+\dim\text{Im}\,f$$

and taking into account that $\;0\le\dim\text{Im}\,f\le 1\;$ ...

In your case, $\;V=\Bbb C^2_{\Bbb C}\;\;,\;\;\Bbb F=\Bbb C\;$ and etc.

Linear maps (complex)

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