Grassmann's Identity/Proof 2

Theorem

Let $K$ be a division ring.

Let $\struct {G, +_G, \circ}_K$ be a $K$-vector space.

Let $M$ and $N$ be finite-dimensional subspaces of $G$.

Then the sum $M + N$ and intersection $M \cap N$ are finite-dimensional, and:

$\map \dim {M + N} + \map \dim {M \cap N} = \map \dim M + \map \dim N$

Proof

By the second isomorphism theorem:

$\dfrac {M + N} M \equiv \dfrac N {M \cap N}$

The result follows.

$\blacksquare$

Source of Name

This entry was named for Hermann Günter Grassmann.