# Find basis to quotient space of 2 spaces

My question is as follows:

V is the space of all n by n matrices. W is a subset of V, and is defined by the space of all symmetric n by n matrices.

We are asked to find a basis for V/W

I don't know where to begin even, I don't understand this material very well and can use some help.

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Start by answering these questions: What is the dimension of $V$? And what is the dimension of $W$? –  Prahlad Vaidyanathan Oct 17 '13 at 14:00
The dimension of V is n squared, and the dimension of W is n*(n+1)/2 –  Oria Gruber Oct 17 '13 at 14:06
Next: What is the dimension of $V/W$? and can you find a basis for $V$? –  Prahlad Vaidyanathan Oct 17 '13 at 14:08
the dimension of V/W is equal to n*(n-1)/2 –  Oria Gruber Oct 17 '13 at 14:10
basis for V, we could use the standard base –  Oria Gruber Oct 17 '13 at 14:11

1. Let $W'$ be the space of skew-symmetric matrices. Show that the map from $W' \to V/W$ given by $$A \mapsto \overline{A}$$ is an isomorphism.
2. Now you need to find a basis for $W'$, so start with a basis for $V$, and note that any $A \in V$ can be written as $$A = \frac{A+A^t}{2} + \frac{A-A^t}{2} \in W + W'$$
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