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I am studying for the exam next week and I am reviewing dimensions.
So far I put together:

  • If $f:V \to W$ is a linear mapping between finite-dimensional vector spaces, then
    \begin{gather*} \dim(V) = \dim(\ker(f)) + \dim(\operatorname{im}(f)), (1)\\ \dim(\hom(V, W)) = \dim(V) \cdot \dim(W). (2) \end{gather*}

  • For linear subspaces $U_1$ and $U_2$: $$ \dim(U_1 + U_2) = \dim(U_1) + \dim(U_2) - \dim(U_1\cap U_2). $$

  • Direct sum $\oplus$:
    $$ \dim(U_1 \oplus U_2) = \dim(U_1) + \dim(U_2). $$

  • Quotient spaces: If $U$ is a subspace of $V$, then $$ \dim(V/U) + \dim(U) = \dim(V). $$

  • Dual space: for a finite dimensional space $V$ (special case of $(2)$) $$ dim(V) = dim(V*) $$

Are there other theorems for dimensions?

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    All this is true for *finite* dimensions, of course... – 2017-02-07
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    If $V$ is finite-dimensional then $f$ is injective iff $dim(im(f))=dim (V).$ (A useful corollary to your first equation.)... If $dim (V)=n$ and $B$ is a linearly independent subset of $V$ there is a set $C$ with $B\subset C\subset V $ where $C$ is linearly independent and has $n$ members.... – 2017-02-07
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    @Chiray: While LaTeXifying the operators, I took the liberty of re-titling your question. If this doesn't match your intent, please revert or otherwise edit. – 2017-02-07
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    @AndrewD.Hwang thank you, I couldnt find the symbol for the direct sum. @ user254665 I didn't know that corollary, so thank you! – 2017-02-07

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In fact, you have more than you need here. The "direct sum" equation is just a special case of the sum equation above it, and the quotient space equation is just a special case of the rank-nullity theorem at the top (take $f : V \to V/U$ to be the projection map).

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    I know, they are all related in some ways, except the one for the hom(V, W), I think. Do you know any other equations? – 2017-02-07