# How to prove that the dim of two subspaces added together equals dim

## Main Question or Discussion Point

How to prove that the dim of two subspaces added together equals dim of their union plus 1 iff one space is a subest of the other

In other words,
subspaces: V, S of Vector space: W

$$dim(V+S) = dim(V \cap S) +1$$

if $$V \subseteq S$$ or $$S \subseteq V$$

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I dont think this is true. If V=S then arent V+S and V intersect S the same?

jbunniii
Homework Helper
Gold Member

eok20 is correct, the statement is false.

What is true is

$$dim(V+S) = dim(V) + dim(S) - dim(V \cap S)$$

which reduces to a simpler form if $V \subseteq S$ or $S \subseteq V$.

I dont think this is true. If V=S then arent V+S and V intersect S the same?
perhaps I mistyped the question.
$$dim(V+S) = dim(V \cap S) +1$$ is given for these subspaces.

One has to prove that

$$V \subseteq S$$ or $$S \subseteq V$$

jbunniii
Homework Helper
Gold Member

perhaps I mistyped the question.
$$dim(V+S) = dim(V \cap S) +1$$ is given for these subspaces.

One has to prove that

$$V \subseteq S$$ or $$S \subseteq V$$
OK, suppose

$$V \not\subseteq S$$ and $$S \not\subseteq V$$.

Then V contains at least one element not in S, hence at least one element not in $V \cap S$. Thus $dim(V) > dim(V \cap S)$, and since dimensions are integers, this is the same as $dim(V) \geq dim(V \cap S) + 1$.

Similarly, S contains at least one element not in V, so $dim(S) \geq dim(V \cap S) + 1$.

Now apply those inequalities to

$$dim(V+S) = dim(V) + dim(S) - dim(V \cap S)$$

How do I come to the fact that

$$dim(V+S) = dim(V) + dim(S) - dim( V \cap S)$$

OK, suppose

$$V \not\subseteq S$$ and $$S \not\subseteq V$$.

Then V contains at least one element not in S, hence at least one element not in $V \cap S$. Thus $dim(V) > dim(V \cap S)$, and since dimensions are integers, this is the same as $dim(V) \geq dim(V \cap S) + 1$.

Similarly, S contains at least one element not in V, so $dim(S) \geq dim(V \cap S) + 1$.

Now apply those inequalities to

$$dim(V+S) = dim(V) + dim(S) - dim(V \cap S)$$

jbunniii
Homework Helper
Gold Member

How do I come to the fact that

$$dim(V+S) = dim(V) + dim(S) - dim( V \cap S)$$
This is a standard result that should be in just about any linear algebra textbook. E.g., theorem 2.18 in Sheldon Axler's "Linear Algebra Done Right." See page 33 in this online preview:

Here is another proof (PDF file), but it looks more longwinded than it needs to be:

http://www.its.caltech.edu/~clyons/ma1b/intsumdimthm.pdf [Broken]