Let V and W be n-dimensional vector spaces, and let T:V-->W be a linear transfor
ID: 2940391 • Letter: L
Question
Let V and W be n-dimensional vector spaces, and let T:V-->W be a linear transformation. Suppose that is a basisfor V. Prove that T is an isomorphism if and only if T( ) isa basis for W. I would greatly appreciate a full proof since this was on aprevious exam by my professor and I have an exam this Friday. Ihave started but I think I am on to something wrong. Pleasecheck back because I will maybe have a question or two. Thanks in advance Let V and W be n-dimensional vector spaces, and let T:V-->W be a linear transformation. Suppose that is a basisfor V. Prove that T is an isomorphism if and only if T( ) isa basis for W. I would greatly appreciate a full proof since this was on aprevious exam by my professor and I have an exam this Friday. Ihave started but I think I am on to something wrong. Pleasecheck back because I will maybe have a question or two. Thanks in advanceExplanation / Answer
To show that a linear map T:V-> W is an isomorphism isequivalent to showing two things - that it is injective, and thatits image is W. For a linear map, it suffices to indicate the images of theelements of a basis, so in particular, it suffices to know theimages of the members of . Proof of the v = 0 and this proves that T is injective. Proof of the => part: Suppose T is an isomorphism and let ={v1,v2 ...,vn} be a basis for V.We need to show that T()={T(v1),T(v2),... T(vn)} is a basis for W. Let uswrite T(vi ) = wi. To show that thewi are a basis we need to show that they are linearlyindependent and that they are also spanning. Suppose 1w1 + 2w2 +... + kwk = 0. Then we may write thisas 1T(v1) +2T(v2) + ... +kT(vk) =T(1v1 + 2v2+ ... + kvk) = 0. Since T is anisomorphism, it follows that 1 v1 + 2v2+ ... + kvk = 0. Since is abasis for V, this in turn implies that all the i =0. But then this implies that the wi are linearlyindependent. To see that they are spanning, consider any w in W. Since T is anisomorphism, it is surjective, so there is a v in V such thatT(v)=w. Since is a basis v can be written as a linearcombination of the vi so let us write v =1v1 +2v2 + ... +kvk. But then w=T(v) = T(1v1 +2v2 + ... +kvk) =1T(v1) +2T(v2) + ... +kT(vk) =1w1 + 2w2 +... + kwk and this proves that the wi span W.Related Questions
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