Linear & Abstract algebra

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10:10 AM

Mar 9 at 12:59, by Martin Sleziak

For instructions how to render MathJax(TeX) in chat see this post on meta.

In response to this:

in Mathematics, 2 hours ago, by Ilan Aizelman WS

Hey guys. Can anyone explain me how Brian did elementary row operations on the matrix here http://math.stackexchange.com/questions/809178/linearly-independence-over-c-and-‌over-r/809239?noredirect=1#809239 I just did it and my lamda1,2,3 were all $/in$ R and I didn't get a complex number as he did. Thx!!

First we can get rid of one of the rows:

$$
\begin{pmatrix}
i & 1 & 1 \\
1 &-i &-i \\
-i & 1 & 0
\end{pmatrix}\sim
\begin{pmatrix}
1 &-i &-i \\
1 &-i &-i \\
-i & 1 & 0
\end{pmatrix}\sim
\begin{pmatrix}
0 & 0 & 0 \\
1 &-i &-i \\
-i & 1 & 0
\end{pmatrix}
$$

Now let us try with the two equations that are left:

$$
\begin{pmatrix}
1 &-i &-i \\
-i & 1 & 0
\end{pmatrix}\sim
\begin{pmatrix}
1 &-i &-i \\
1 & i & 0
\end{pmatrix}\sim
\begin{pmatrix}
1 &-i &-i \\
2 & 0 &-i
\end{pmatrix}\sim
\begin{pmatrix}
2 & 0 &-i \\
2 &-2i&-2i
\end{pmatrix}\sim
\begin{pmatrix}
2 & 0 &-i \\
0 &-2i&-i
\end{pmatrix}\sim
\begin{pmatrix}
2 & 0 &-i \\
0 & 2 & 1
\end{pmatrix}\sim
$$

This leads to the solution $(i,-1,2)$.

I am not sure, but things like this might also be solvable with WolframAlpha. At least for some types of problems there is an option to "show steps" on the WA website.

8 hours later…