[Linear Algebra] Help with Linear Transformations part 2

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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.
 

Answers and Replies

  • #3
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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.
 
  • #4
mathwonk
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check out the relatively prime decomposition theorem on page 26 of these notes;

http://alpha.math.uga.edu/~roy/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/~roy/laprimexp.pdf
 

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