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this was a hw question given in class today, but I am not sure where to begin the proof. There are so many theorems that we went over today, I'm not sure which ones are applicable, and which ones to use. If anyone could help me out, that would be great.

Question:

Let S be a subspace of a finite dimensional vector space V and suppose that there are subspaces $T_1$ and $T_2$ such that $V = S \oplus T_1$ and $V = S \oplus T_2.$

Prove that $\dim(T_1) = \dim(T_2)$.

Surb
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MathWeep
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    Take a basis of $B_S$ of $S$, $B_1$ of $T_1$ and $B_2$ of $T_2$. What can you say about the sets ${ B_S,B_1}$ and ${B_S,B_2}$ ? – Surb Oct 23 '18 at 21:04

2 Answers2

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$$\dim(V)=\dim(S)+\dim(T_1)=\dim(S)+\dim(T_2) \implies \dim(T_1)=\dim(T_2)$$

Surb
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keunungPark
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Proof Idea: Take a basis $B_S$ of $S$, $B_1$ of $T_1$ and $B_2$ of $T_2$. The fact that $S\oplus T_i = V$ is a direct sum allows you to show that the set $\{ B_S,B_i\}$ is a basis of $V$ for $i=1,2$. As $V$ is finite dimensional, you can deduce that $|\{ B_S,B_1\}|=|\{B_S,B_2\}|$ and $|\{B_S,B_i\}|=|B_S|+|B_i|$.

I leave it to you to justify carefully each step.

Surb
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