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$\begingroup$

Finding the basis for the kernel of:

\begin{pmatrix} a & b \\c & d\end{pmatrix}

$which$ $maps$ $to:$

\begin{pmatrix} a \\a\\3a + b \end{pmatrix}

It's all complex, but I'm not sure if that's relevant!

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    I think it is about time you go to the FAQ section and read about how to use LaTeX to properly write mathematics in this site.2012-11-28
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    I can figure out how you (probably) intended the matrix to be formatted, but what do you mean by "C^3(a, a, 3a+d)"? As DonAntonio points out, we won't run into such troubles if you format the posts, yourself.2012-11-28
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    Honest: try to get out of that hectic rythm for some minutes and learn how to type properly mathematics, so that people won't have to guess what you meant. The time of us all is important.2012-11-28
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    @DonAntonio I've given it a go for you ;)2012-11-28
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    Many thanks, @Becky...but I really think you've given a go *for yourself*. :)2012-11-28
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    Alright Don if you're going to be pedantic! :P2012-11-29

1 Answers 1

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So you have the map $F:\mathbb C^{2\times 2} \longrightarrow \mathbb C^3, \ \begin{pmatrix}a&b\\c&d \end{pmatrix} \mapsto\begin{pmatrix}a\\a\\3a+b \end{pmatrix}$ and you want to find $\ker F$.
By definition $$\ker F=\left\{ \begin{pmatrix}a&b\\c&d \end{pmatrix}\in\mathbb C^{2\times 2}:F\left(\begin{pmatrix}a&b\\c&d \end{pmatrix}\right)=\begin{pmatrix}0\\0\\0 \end{pmatrix}\right\}\\ =\left\{ \begin{pmatrix}a&b\\c&d \end{pmatrix}\in\mathbb C^{2\times 2}:\begin{pmatrix}a\\a\\3a+b \end{pmatrix}=\begin{pmatrix}0\\0\\0 \end{pmatrix}\right\}\\ =\left\{ \begin{pmatrix}a&b\\c&d \end{pmatrix}\in\mathbb C^{2\times 2}:a=0,3a+b=0\right\}=\cdots$$ After that find the basis.

An answer is

$\left\{\begin{pmatrix}0&0\\1&0 \end{pmatrix},\begin{pmatrix}0&0\\0&1 \end{pmatrix}\right\}$

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    So is the basis \begin{pmatrix} 0 & 0 \\r & s\end{pmatrix} ? Where r and s are just constants which don't affect it?2012-11-28
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    The basis must contain $2$ elements. The $\left\{\begin{pmatrix}0&0\\ r & s \end{pmatrix} \right\}$ is not a basis for any $r,s$. The $\left\{\begin{pmatrix}0&0\\ r&0\end{pmatrix}, \ \begin{pmatrix}0&0 \\ 0 & s \end{pmatrix}\right\}$ is a basis for every $r,s\neq0$.2012-11-28
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    Oh sorry, I meant that's the kernel.2012-11-28