Show that two linear maps are equal if they agree on a basis

[mrroboto]

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"?

 

[mrroboto]

Homework Statement

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.

Homework Equations

none

The Attempt at a Solution

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"?

 

[Dick]

Maps (in L(V,W)) don't have bases. Vector spaces have bases. Can you rephrase the question knowing that?

 

[morphism]

(I suppose you also meant T is in L(V,W) as well.)

What do you mean when you say "S has a basis"?! S is a linear map, not a vector space. V is what has the basis {v_1, ..., v_n}. Now to show that S=T, it will be enough to show that T(v)=S(v) for all v in V. But each v in V has a unique representation in terms of the basis vectors. So use this (and don't forget that T and S are linear).

 

[HallsofIvy]

Homework Statement

Suppose V and W are vector spaces, and {v1...vn} is basis for V and T. S is an element of L(V, W).

?? You mean, I think, that "{v1,..., vn} is a basis for V." and that "T and S are elements of L(V,W).

Suppose further that T(vi)=S(vi) for all i with 1<= i <=n. Show that S=T.

Homework Equations

none

The Attempt at a Solution

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"?

As Dick pointed out, linear maps do not have a basis! You mean that {v1,...,vn} is a basis for V (which you had already said). In order to prove that T= S, you must prove that T(v)= S(v) for any vector v in V. v can be written as a linear combination of the basis vectors: v= a1v1+ ...+ a2v2. What happens if you apply both T and S to that?

 

[HallsofIvy]

This was also posted in the "homework" area so I am merging the two threads.

mrroboto, do NOT double post!