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.(adsbygoogle = window.adsbygoogle || []).push({});

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"?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Vector spaces

**Physics Forums | Science Articles, Homework Help, Discussion**