# how to guess dimensions in the following subspaces

let $S$ and $T$ be two subspaces of $R^{24}$, such that $\dim(S)= 19$ and $\dim(T)= 17$, then the

a. Smallest possible value of $\dim(S \cap T)$ is ?

b. largest possible value of $\dim(S \cap T)$ is ?

c. Smallest possible value of $\dim(S + T)$ is ?

d. largest possible value of $\dim (S + T)$ is ?

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What have you tried? Have you heard of the dimension formula? –  chris Sep 30 '12 at 16:37
If not, the dimension formula is $\dim(S+T)=\dim(S)+\dim(T)-\dim(S\cap T)$. It'll help to think about the case when $T\subset S$ as well as the case when $S$ and $T$ jointly span $\mathbb{R}^{24}$. –  Kevin Carlson Sep 30 '12 at 16:39
ok sir!! I am trying to solve it. –  ram Sep 30 '12 at 16:41
I changed $dim(S)$ to $\dim(S)$, coded as \dim(S). These doesn't only prevent italicization, but also results in proper spacing in things like $a\dim B$. –  Michael Hardy Sep 30 '12 at 17:16

$$(1)\;\;\;\;\;\;\dim(S+T)=\dim S+\dim T-\dim(S\cap T)$$
$$(2)\;\;\;\;\;\;\forall\;\; K\leq \Bbb R^{24}\,\,,\,\,0\leq\dim K\leq \dim\Bbb R^{24}=24$$