[Linear Algebra] Help with Linear Transformations part 2

[iJake]

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.

 

[fresh_42]

See my comment in your first thread.
https://www.physicsforums.com/threa...transformation-exercises.948632/#post-6006630

 

[iJake]

I apologize for taking so long to reply to this thread! I was away.

I'm still finding this one a bit tricky. At first I took $u \in S$ and $w \in A$ but these were not working for me I don't think. Now I've reached

$u \in A = j(v) + v = -v + v$
$w \in S = j(v) - v = v - v$

$v = u + w = (j(v) + v) + (j(v) - v) = (-v + v) + (v - v) = 0 + 0 = 0$

Is this the idea I'm meant to apply? I'm dubious as I don't think I really used the fact that $j \circ j = id_V$.

I suppose once I'm sure about the methodology for this first part I'll be better equipped to deduce the relationship between symmetric and skew-symmetric matrices. What does my problem mean when it says to find the convenient linear map $j$?

Thanks again for taking the time to assist me.

 

[mathwonk]

check out the relatively prime decomposition theorem on page 26 of these notes;

http://alpha.math.uga.edu/%7Eroy/4050sum08.pdf


it says that since your map satisfies the polynomial X^2-1 = (X-1)(X+1), then your space is a direct sum of subspaces on which your map satisfies ether X-1 or X+1.

it is not entirely trivial since the proof uses the euclidean algorithm as i recall.

or look at the equivalent decomposition lemma at the beginning of chapter 4 of these notes:

http://alpha.math.uga.edu/%7Eroy/laprimexp.pdf