- #1

- 70

- 0

## Homework Statement

A subspace N of a vector space V has finite codimension n if the quotient space V/N is finite-dimensional with dimension n. Show that a subspace N has finite codimension n iff N has a complementary subspace M of dimension n. Do not assume V to be finite-dimensional.

**2. The attempt at a solution**

Let [tex]\left\{N+\alpha_i \right\}[/tex] ([tex]1\leq i \leq n[/tex]) be the basis of V/N, I want to show the set spanned by [tex]\alpha_i[/tex] is the complementary subspace M.

First I show V=N+M:

since [tex]\left\{N+\alpha_i \right\}[/tex] are the basis, each v in V can be represented as [tex]\eta+\sum x_i \alpha_i, \eta \in N[/tex]

Next I prove [tex]N\bigcap M[/tex] = {0}:

if this is not the case, there must be some element in N that can be represented as [tex]\sum x_i \alpha_i[/tex]. Since N is a subspace, this means [tex]\alpha_i[/tex] must be in N. Therefore, [tex]\left\{N+\alpha_i \right\}[/tex] cannot be a basis for V/N

Am I correct?

Thanks