Suppose $U$ is a finite dimensional vector space and $U$ can be expressed as a direct sum $U=U_1\oplus\dots\oplus U_n$. Then $\dim(U) = \dim(U_1) + \dots + \dim(U_n)$.
I don't really have any idea how to prove this, any tips?
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1What are you using as the definition of $U_1\oplus U_2$? – Brian Fitzpatrick May 10 '15 at 22:00
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Searching for "dimension of a direct sum" brings up a number of relevant results. – May 11 '15 at 01:47
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Hint. A general theorem states that $$ \dim(U_1+U_2)=\dim(U_1)+\dim(U_2)-\dim(U_1\cap U_2)\tag{1} $$ Can you prove this?
Once we have this general theorem, note that $V=U_1\oplus U_2$ means $V=U_1+U_2$ and $U_1\cap U_2=\{\vec 0\}$. Thus the formula $$ \dim(U_1\oplus U_2)=\dim(U_1)+\dim(U_2) $$ is a corollary of the formula in (1). Your general formula should then follow from induction.
Brian Fitzpatrick
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Ah yes that makes sense, $\dim{(U_1\cap U_2)} = 0$, thus $\dim{(U_1+U_2)} = \dim{(U_1)}+\dim{(U_2)}$. Then I understand the induction, thanks. – user2850514 May 11 '15 at 00:27