- #1

zwingtip

- 20

- 0

## Homework Statement

let [tex]V[/tex] be a finite dimensional vector space of dimension n. For [tex]W \leq V [/tex] define the codimension of [tex]W[/tex] in [tex]V[/tex] to be [tex]codim(W) = dim(V) - dim(W)[/tex]. Let [tex]W_i, 1 \leq i \leq r[/tex] be subspaces of [tex]V[/tex] and [tex]S = \cap_{i=1}^{r}W_i[/tex]. Prove:

[tex]codim(S) \leq \sum_{i=1}^{r} codim(W_i)[/tex]

## Homework Equations

[tex] dim(U + V) = dim(U) + dim(V) - dim(U \cap V)

\sum_{i=1}^{r} codim(W_i) = \sum_{i=1}^{r} (n - dim(W_i)[/tex]

## The Attempt at a Solution

I'm completely lost here. I know I need to prove this by induction. Any tips to point me in the right direction? Can I use the fact that [tex]codim(S) = n - dim(\cap_{i=1}^{r} W_i[/tex] and [tex] dim(U \cap V) < dim(U) + dim(V)[/tex] assuming [tex]dim(U + V) \neq 0[/tex]?