- #1
Ninty64
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Homework Statement
This question came out of a section on Correspondence and Isomorphism Theorems
Let [itex]V[/itex] be a vector space and [itex]U \neq V, \left\{ \vec{0} \right\} [/itex] be a subspace of [itex]V[/itex]. Assume [itex]T \in L(V,V)[/itex] satisfies the following:
a) [itex]T(\vec{u} ) = \vec{u}[/itex] for all [itex]\vec{u} \in U[/itex]
b) [itex]T(\vec{v} + U) = \vec{v} + U[/itex] for all [itex]\vec{v} \in V[/itex]
Set [itex]S=T-I_{V}[/itex]. Prove that [itex]S^{2}=\vec{0}_{V \rightarrow V}[/itex]
Homework Equations
[itex]I_{V}[/itex] is the identity map
[itex]L(V,V)[/itex] is the map of all linear operators on V
The Attempt at a Solution
I have trouble understanding the question.
Since [itex]T \in L(V,V)[/itex] then how is [itex]T(\vec{v} + U) = \vec{v} + U[/itex] for all [itex]\vec{v} \in V[/itex]?
Wouldn't that mean [itex]T:V/U \rightarrow V/U[/itex]?
I don't understand, what is [itex]T(\vec{v})[/itex] equal to?
Does [itex]T(\vec{v})=\vec{v}[/itex] or [itex]T(\vec{v}) = [\vec{v}]_W[/itex] or something else?
I'm sorry if this is a silly question.