The composition of linear maps and their underlying spaces.
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Let U, V, and W be vector spaces over a field F. Suppose that T : U --> V and S : V --> W are linear transformations and that Im(T) = Ker(S). If T is injective and S is surjective, prove that
dim(V) = dim(U) + dim(W).
Here, dim denotes the dimension of a vector space over the field F.
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Solution Summary
The rank-nullity theorem is used to derive a relationship between the vector spaces involved in the composition of an injective and surjective linear map. The step-by-step solution to this problem illustrates the use of fundamental concepts in linear algebra.
Solution Preview
For ease of notation I will drop reference to the field F. We assume throughout that F is fixed and that dim refers to dimension of a vector space over F. This problem boils down to the following fundamental property of linear transformations, called the rank-nullity theorem:
Let X and Y be vector spaces and let L : X --> Y be a ...
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