- #1
p3forlife
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Homework Statement
Let V be a vector space over a field F and let L(V) be the vector space of linear transformations from V to V. Suppose that T is in L(V). Do not assume that V is finite-dimensional.
a) Prove that T^2 = -T iff T(x) = -x for all x in R(T).
b) Suppose that T^2 = -T. Prove that the intersection of N(T) and R(T) = {0}.
Homework Equations
The Attempt at a Solution
The T^2 is throwing me off slightly. Does it just mean take the square of the original linear transformation T?
So if T^2 = -T,
T^2 (x) = -T(x)
Also rank(T^2) = rank (-T)
That's all I have so far. How can I approach this if I can't assume that V is finite-dimensional?
If I could get a few tips on how to start this question, it would be very helpful :)