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Let $U$ be upper triangular matrices in $M_2$ and $L$ lower triangular matrices in $M_2$. Show that their sum isn't direct, and that their sum is the whole vector space.

I have the following definitions:

If $L\cap M=\{0\}$, then we say that that is the direct sum. Notation: $L\oplus M$.

If $L\oplus M=V$, we say that $M$ is a direct complement for $L$ and vice versa.

I know that an upper triangular matrix can be written as $\begin{pmatrix} a & b\\0 & d\end{pmatrix}$ and lower triangular as $\begin{pmatrix} 0 & 0\\c & 0\end{pmatrix}$. If I add them up, I get $\begin{pmatrix} a &b \\c & d\end{pmatrix}$. But how do I prove their sum is the whole space $M_2$? And what about direct sum?

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Hint:

In $M_2(\mathbb{K})$, Lower triangular matrices have the form: $$ \begin {pmatrix} m&0\\ n&q \end{pmatrix} $$ so the intersection of $L$ and $M$ is the set of all diagonal matrices (and this answer to your first question).

For the second question, show that any matrix of the form $$ \begin {pmatrix} x&y\\ z&t \end{pmatrix} $$ can be expressed as a sum: $$ \begin {pmatrix} x&y\\ z&t \end{pmatrix} = \begin {pmatrix} 0&0\\ z&0 \end{pmatrix}+ \begin {pmatrix} x&y\\ 0&t \end{pmatrix} $$ where the two matrices are elements of $L$ and $M$.

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  • $\begingroup$ Why do I have to show that any matrix in $M_2$ can be expressed as a linear combination of elements in $L$ and $U$? $\endgroup$ – lmc Nov 17 '16 at 10:09
  • $\begingroup$ Sorry. My answer was not clear. I've edit and I hope that now it's better. $\endgroup$ – Emilio Novati Nov 17 '16 at 10:22

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