12-240/Proofs in Vector Spaces
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Theorems & Proofs
Theorem: Let be a subspace of a finite dimensional vector space . Then is finite dimensional and
Proof: Let be a basis for . Then we know that is a finite set since is a finite dimensional. Then, for given a subspace , let us construct a linearly independent set by adding vectors from such that is maximally linearly independent. In other words, adding any other vector from would make linearly dependent. Here, L has to be a finite set by the Replacement Theorem, if we choose the generating set as , so since is a some linearly independent subset of . Now we want to show that is a basis for . Since is linearly independent, it suffices to show that . Suppose not:. (We know that since is made of vectors from .) Then But this means is linearly independent, which contradicts with maximally linearly independence of . Therefore and hence, is a basis for
Replacement Theorem: Let be a vector space generated by (perhaps linearly dependent) where and let be a linearly independent subset of such that . Then and there exists a subset of with and .
Proof: We will prove by induction hypothesis on :
For : , and so,
Now, suppose true for :