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## Homework Statement

## Homework Statement

(a) Let ##V## be an ##\mathbb R##-vector space and ##j : V \rightarrow V## a linear transformation such that ##j \circ j = id_V##. Now, let

##S = \{v \in V : j(v) = v\}## and ##A = \{v \in V : j(v) = -v\}##

Prove that ##S## and ##A## are subspaces and that ##V = S \oplus A##.

(b) Deduce that the decomposition of the matrices in direct sum from the symmetric and skew-symmetric matrices from part (a) (finding a convenient linear transformation ##j##)

[apologies if that last part is a bit weird sounding, I'm translating from Spanish]

## Homework Equations

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## The Attempt at a Solution

a) The test for ##S## and ##A## being subspaces is fairly trivial so I don't include it. Now, to determine that ##V = S \oplus A## I'm finding it a little trickier. I observe clearly that ##S \cap A = \{0\}## but how do I formalize that and lead into it proving that V is the direct sum of S and A?

b) is also confusing me. I found this and it looks remarkably similar to my problem, but I do not know how to apply it here.

Thank you Physics Forums for any help.