Dimensional formula

The dimensional formula comes from the mathematical branch of linear algebra . It indicates how the dimension of the sum of two finite-dimensional sub-vector spaces , a larger vector space, can be calculated: ${\displaystyle V_{1}}$ ${\displaystyle V_{2}}$

${\displaystyle \dim \left(V_{1}+V_{2}\right)=\dim V_{1}+\dim V_{2}-\dim \left(V_{1}\cap V_{2}\right)}$

It follows directly from the ranking . The situation is a special case (see direct sum ). In this case the dimensional formula is reduced to ${\displaystyle V_{1}\oplus V_{2}=V_{1}+V_{2}}$

${\displaystyle \dim \left(V_{1}+V_{2}\right)=\dim V_{1}+\dim V_{2},}$

since applies to a direct sum

${\displaystyle V_{1}\cap V_{2}=\{0\}.}$

The sub-vector space represented by the intersection of and is thus the zero vector space , the dimension of which is equal to zero. ${\displaystyle V_{1}}$ ${\displaystyle V_{2}}$

If or is infinite dimensional, it is no longer possible to carry out the subtraction . However, it applies in any case ${\displaystyle V_{1}}$ ${\displaystyle V_{2}}$

${\displaystyle \dim \left(V_{1}+V_{2}\right)\geq \max\{\dim V_{1},\dim V_{2}\}}$

and

${\displaystyle \dim \left(V_{1}+V_{2}\right)\leq \dim \left(V_{1}\oplus V_{2}\right)=\dim V_{1}+\dim V_{2}}$.

Since for two cardinal numbers , of which at least one endless , the sum is equal to the maximum of the two, that is, in the case that one of the two partial spaces is infinite dimensional, . ${\displaystyle \dim \left(V_{1}+V_{2}\right)=\max\{\dim V_{1},\dim V_{2}\}}$

literature

• Siegfried Bosch : Linear Algebra. Springer-Verlag, 2001, ISBN 3-540-41853-9 , pp. 46-47.