Show that the sum of a set of matrices isn't direct, and that the sum is the whole vector space.

Question

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?

Answer 1

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 $M$.

Answer 2

I think the answer above is incomplete, and I'd like to complete it. In order for $M_2(\mathbb{K})$ to not be equal to the direct sum, this is, for $M_2(\mathbb{K})$ to differ from $U \bigoplus L$ we need to know the definition of direct sum which says:

each $x$ in $F_1 + ... + F_n$ is written in an unique way as the sum $x = x_1 + ... + x_n $

which is equivalent to say:

for each $j = 1, ..., k$ we have $F_j \bigcap (F_1 + ... + F_{j-1} + F_{j+1}+...+ F_n) = \{0\}$

and it's evident that $M_2(\mathbb{K})$ isn't written as a superior and inferior triangular matrices in an unique way. That's why, we say that the direct sum is different from $M_2$.

Source: Alégra Linear, Elon Lages (a brazilian mathematician).