How to find the basis in the space of the following matrices?

Question

I'm struggling with understanding Linear Algebra. Sometimes I find problems like this:

Find a basis for each of these subspaces of 3 by 3 matrices:

1. All diagonal matrices
2. All symmetric matrices ($A^T = A$)
3. All skew-symmetric matrices ($A^T = -A$)

And then I just have minimal ideas of how to find a solution that can fit a set of problems of the same type - just like on the picture(all diagonal matrices, all symmetric ones, and so on). Let's take for example a second problem. I would suggest to make a custom symmetric matrix and find a basis for it and after that derive a solution. However, this approach does not work well for me. May be someone has faced something like this and have successfully found a good way of thinking in such abstractions?

Answer 1

Writing down a matrix of the given kind that is as arbitrary as you can is a very good approach:

• Consider the amount of parameters that you need to do that, this is most likely going to be the dimension of the subspace.
• Use linearity to split your arbitrary matrix into a sum where each parameter is involved only once in each term: this exposes the desired basis.

E.g. an arbitrary $2\times 2$ matrix can be written uniquely $$\left(\matrix{a& b\\c &d}\right)$$ You can see $4$ parameters, and you can rewrite it as $$a\left(\matrix{1& 0\\0 &0}\right)+b\left(\matrix{0& 1\\0 &0}\right)+c\left(\matrix{0& 0\\1 &0}\right)+d\left(\matrix{0& 0\\0 &1}\right)$$ where you can see your basis.

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The fact that any matrix $\color{red}{\text{can be written}}$ $\color{blue}{\text{uniquely}}$ as a linear combination of these four proves that they form a $\color{blue}{\text{linearly independent}}$ $\color{red}{\text{spanning set}}$, aka. a basis.