Basis for upper triangular matrices of $M_n(F)$
I'm asked to find a basis for $W$, which is a subspace of $M_n(F)$.
$W$ is the subspace containing all upper triangular $n \times n$ matrices.
How do you find this basis?My guess is that it's simply a collection of $[n(n+1)]/2$ matrices $E^{ij}$ in which $ij = 0: i>j$.
Would that be correct, and if so, is there a better way to express it? How do I show that it spans?
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$\begingroup$Your answer is indeed correct.
Showing both that this set spans all upper triangular matrices and showing that it is linear independent should be very easy (but if you are having problems with any feel free to update your question so I could add details to this answer)
Added: To show that this set spans all upper triangular matrices, take an upper triangular matrix. if the $(i,j)$ co-ordinate is $a$ what would you take as the coefficient of $E_{i,j}$ ? Can the other matrices in the sum affect this coordinate ?
Example: $$\begin{pmatrix}a & b\\ 0 & c \end{pmatrix}=a\cdot\begin{pmatrix}1 & 0\\ 0 & 0 \end{pmatrix}+b\cdot\begin{pmatrix}0 & 1\\ 0 & 0 \end{pmatrix}+c\cdot\begin{pmatrix}0 & 0\\ 0 & 1 \end{pmatrix}$$
Can you generelize for $n$ ?
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