Grassmann's Identity

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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 1

Outline

Starting from a basis of $M \cap N$, we complete it to a basis of $M$ and one of $N$.

We then verify that the union of these basis is a basis of $M + N$.


Proof

First, suppose $M \subseteq N$ or $N \subseteq M$.

Then the assertion is clear.


Assume that $M \cap N$ is a proper subspace of both $M$ and $N$.

Let $B$ be a basis of $M \cap N$.

By Dimension of Proper Subspace is Less Than its Superspace this is finite-dimensional.

By Results concerning Generators and Bases of Vector Spaces, there exist non-empty sets $C$ and $D$ disjoint from $B$ such that:

$B \cup C$ is a basis of $M$
$B \cup D$ is a basis of $N$.

The space generated by $B \cup C \cup D$ contains both $M$ and $N$.

Hence it contains $M + N$.

But as $B \cup C \cup D \subseteq M \cup N$, the space it generates is contained in $M + N$.

Therefore $B \cup C \cup D$ is a generator for $M + N$.


If $d$ is a linear combination of $D$ and also of $B \cup C$, then $d \in M \cap N$.

So $d$ is a linear combination of $B$, and consequently $d = 0$ as $B \cup D$ is linearly independent and $D$ is disjoint from $B$.

In particular, $D$ is disjoint from $B \cup C$.

$\Box$


Next we show that $B \cup C \cup D$ is linearly independent and hence a basis of $M + N$.

Let $\sequence {b_m}$ and $\sequence {d_p}$ be sequences of distinct vectors such that $B \cup C = \set {b_1, \ldots, b_m}$ and $D = \set {d_1, \ldots, d_p}$.

Let $\ds \sum_{j \mathop = 1}^m \lambda_j b_j + \sum_{k \mathop = 1}^p \mu_k d_k = 0$.

Then:

$\ds \sum_{k \mathop = 1}^p \mu_k d_k = - \sum_{j \mathop = 1}^m \lambda_j b_j$

Hence $\ds \sum_{k \mathop = 1}^p \mu_k d_k$ is a linear combination of $D$ and also of $B \cup C$.


By the preceding:

$\ds \sum_{k \mathop = 1}^p \mu_k d_k = 0$

Hence $\mu_k = 0$ for all $k \in \closedint 1 p$.

Thus:

$\ds \sum_{j \mathop = 1}^m \lambda_j b_j = 0$

and therefore $\lambda_j = 0$ for all $j \in \closedint 1 m$.

Therefore $B \cup C \cup D$ is linearly independent.


Thus we have:

\(\ds \map \dim {M + N}\) \(=\) \(\ds \card {B \cup C \cup D}\)
\(\ds \) \(=\) \(\ds \card {B \cup C} + \card D\)
\(\ds \) \(=\) \(\ds \card {B \cup C} + \card {B \cup D} - \card B\)
\(\ds \) \(=\) \(\ds \map \dim M + \map \dim N - \map \dim {M \cap N}\)

Hence the result.

$\blacksquare$


Proof 2

By the second isomorphism theorem:

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

The result follows.

$\blacksquare$


Also known as

This result can also be referred to as Grassmann's formula.


Source of Name

This entry was named for Hermann Günter Grassmann.