12-240/Proofs in Vector Spaces

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...generating set as $\beta$, so $k=|L|\leq |\beta |=dimV$ ***(explanation needed, why? [or your question])*** since $L$ is a some linearly independent...

Theorems & Proofs

Theorem: Let $W$ be a subspace of a finite dimensional vector space $V$. Then $W$ is finite dimensional and $dimW\leq dimV$

Proof: Let $\beta$ be a basis for $V$. Then we know that $\beta$ is a finite set since $V$ is a finite dimensional. Then, for given a subspace $W$, let us construct a linearly independent set $L$ by adding vectors from $W$ such that $L=\{w_{1},w_{2},...w_{k}\}$ is maximally linearly independent. In other words, adding any other vector from $W$ would make $L$ linearly dependent. Here, L has to be a finite set by the Replacement Theorem, if we choose the generating set as $\beta$, so $k=|L|\leq |\beta |=dimV$ since $L$ is a some linearly independent subset of $V$. Now we want to show that $L$ is a basis for $W$. Since $L$ is linearly independent, it suffices to show that $span(L)=W$. Suppose not:$span(L)\neq W$. (We know that $L\subseteq span(L)\subseteq W$ since $L$ is made of vectors from $W$.) Then $\exists w_{a}\in W:w_{a}\notin span(L)$ But this means $span(L)\cup \{w_{a}\}$ is linearly independent, which contradicts with maximally linearly independence of $L$. Therefore $span(L)=W$ and hence, $L$ is a basis for $W$

Replacement Theorem: Let $V$ be a vector space generated by $G$ (perhaps linearly dependent) where $|G|=n$ and let $L$ be a linearly independent subset of $V$ such that $|L|=m$. Then $m\leq n$ and there exists a subset $H$ of $G$ with $|H|=n-m$ and $span(H\cup L)=V$.

Proof: We will prove by induction hypothesis on $m=|L|$:

For $m=0$: $L=\emptyset$, $0\leq n$ and $H=G$ so, $span(H\cup L)=span(H)=span(G)=V$

Now, suppose true for $m$: