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

gutnedawg
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 $$S \subseteq V$$

eok20

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

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 $S \subseteq V$.

gutnedawg

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 $$S \subseteq V$$

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

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

gutnedawg

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

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

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: