# 12-240/Proofs in Vector Spaces

## Important Note About This Page

This page is intended for sharing/clarifying proofs. Here, you might add a proof, correct a proof, or request more detailed explanation of some specific parts of given proofs. To request an explanation for a proof, you may put a sign at that specific part by editing this page. For example:

...generating set as , so *****(explanation needed, why? [or your question])***** since is a some linearly independent...

## 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 :