Grassmann's Identity/Proof 2
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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.