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