Dimensions of vector spaces of $n×n$ symmetric matrix

Question

My book asks for the dimensions of the vector spaces for the following two cases:

1)vector space of all upper triangular $n × n$ matrices, and

2)vector space of all symmetric $n × n$ matrices

The answer for both is $n(n+1)/2$ and this is easy enough to verify with arbitrary instances but what is the formal way to conclude this in the general case, as per the question?

Thanks!

EDIT: I understand how bases and dimensions and linear independe relate. I'm asking how to find the number of dimensions for those two cases above WITHOUT just counting examples where n is equal to, say, 4, 5, 6...

How many elements can be different from zero in the first row? And in the second? ... Add them to get an upper bound for the dimension. Isn't it $n(n+1)/2?$ Can you find $n(n+1)/2$ linearly independent matrices? I bet you can. — mfl, Dec 23 '16 at 13:39
I don't understand the question. "this is easy enough to verify with arbitrary instances (do you mean arbitrary $n$?)" — it seems that you have already had the proof. What is the question then? — , Dec 23 '16 at 14:04
Possible duplicate of Find a basis for the vector space of symmetric matrices with an order of $n \times n$ — , Dec 23 '16 at 14:06
So then is the main difficulty proving that, e.g. the number of entries in the upper half of an $n\times n$ matrix is $n(n+1)/2$? — Zach Boyd, Dec 24 '16 at 02:30

score 1 · Answer 1 · answered Dec 23 '16 at 15:29

One way to formally prove that a vector space has a given dimension is to exhibit a basis and count the number of elements. In fact, the number of basis elements is one definition of dimension.

So, for instance, in the case of symmetric matrices, let $E_{ij}$ be the matrix which is all zeros except for the $i,j$ and $j,i$ entries, which are one. The set $S=\{ E_{ij}, i=1,...,n, i\leq j\}$ is a basis for the space, which can be shown by observing that an arbitrary symmetric matrix $A=(a_{ij})$ can be written as $\sum_{i\leq j} a_{ij} E_{ij}$ and that this is the unique decomposition of $A$ as a linear combination of the $E_{ij}$.

Dimensions of vector spaces of $n×n$ symmetric matrix

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