Upper trianglar matrix is a subspace of mxn matrices

[Mr Davis 97]

Homework Statement

Prove that the upper triangular matrices form a subspace of $\mathbb{M}_{m \times n}$ over a field $\mathbb{F}$

Homework Equations

The Attempt at a Solution

We can prove this entrywise.

1) Obviously the zero matrix is an upper triangular matrix, because it satisfies the property that whenever $i > j$, $A_{ij} = 0$.

2) There are two cases; when $i \le j$ and when $i > j$.
When $i \le j$ we have that $(A + B)_{ij} = A_{ij} + B_{ij}$. When $i > j$ we have that $(A + B)_{ij} = A_{ij} + B_{ij} = 0 + 0 = 0$. Therefore, $A + B$ is still an upper triangular matrix.

3) When $i \le j$ we have that $(cA)_{ij} = c(A_{ij})$. When $i > j$, we have that $(cA)_{ij} = c(A_{ij}) = c(0) = 0$. Therefore, $cA$ is still an upper triangular matrix.

Is this argument enough to show that the upper triangular matrices form a subspace of mxn matrices over a field F?

 

[andrewkirk]

Yes.