Hi-Outlines of the proof.

1.

This is easy to show.

2. So if we show that dimension of LHS = dimension of RHS we are done

3.

This can again be shown easily

4. dim( ) + dim( ) = dim( ) =

We will use this result directly. This is applicable for any sub-space W in V.

5. dim( ) = dim( ) + dim( ) - dim( )

We will use this result directly. This is applicable for any sub-paces S,W in V.

Let dim( ) = s, dim( )=w, dim( ) = i, dim( ) = u

so,

dim( ) = dim( )+dim( ) - dim( )

=

=

= dim( )

Hence we are done.