- #1
mrroboto
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Suppose V and W are vector spaces, and {v1...vn} is basis for V and T. S is an element of L(V, W). Suppose further that T(vi)=S(vi) for all i with 1<= i <=n. Show that S=T.
Here's what I think.
Because S is an element of L(V,W), S:V-->W means that S has a basis of {v1...vn}, and two vector spaces that form a bijective linear map (which S and T do because they have the same basis) are isomorphic. Moreover, because T(vi)=S(vi) then by their isomorphism, T and S must be equal.
This is the last question on my practice midterm and I'm unsure if this is how to proceed with the proof. Any comments, especially on how I could do this more "formally"?
Here's what I think.
Because S is an element of L(V,W), S:V-->W means that S has a basis of {v1...vn}, and two vector spaces that form a bijective linear map (which S and T do because they have the same basis) are isomorphic. Moreover, because T(vi)=S(vi) then by their isomorphism, T and S must be equal.
This is the last question on my practice midterm and I'm unsure if this is how to proceed with the proof. Any comments, especially on how I could do this more "formally"?