The restriction of a linear mapping
If I consider the following paragraph from Greub, Linear Algebra, 1975:
and take the general definition of restriction for a map as already given,
would the following Lemma reproduce the idea of what Greub is saying, or otherwise, what does Greub want to show with this paragraph?
Lemma Given a linear map ϕ:V→W and two subspaces V1⊂V and W1⊂W such that v∈V1⟹ϕ(v)∈W1, then the following diagram commutes:
where ϕ∣V1 is the restriction of ϕ to V1 and iV,iW are canonical injections.
Proof We have to prove that ϕ∘iV1=iW1∘ϕ∣V1. ⋯
EDIT 1 I corrected the Lemma: I had copied the wrong one.
and take the general definition of restriction for a map as already given,
would the following Lemma reproduce the idea of what Greub is saying, or otherwise, what does Greub want to show with this paragraph?
Lemma Given a linear map ϕ:V→W and two subspaces V1⊂V and W1⊂W such that v∈V1⟹ϕ(v)∈W1, then the following diagram commutes:
where ϕ∣V1 is the restriction of ϕ to V1 and iV,iW are canonical injections.
Proof We have to prove that ϕ∘iV1=iW1∘ϕ∣V1. ⋯
EDIT 1 I corrected the Lemma: I had copied the wrong one.
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