Commutative

The dot product of column matrices is commutative:

a · b  =  b · a

As with the dot product of vectors, the order of operands makes no difference. Write out the definition of dot product for both arrangements of a and b:

a · b  =  ( a₁, a₂ )^T · ( b₁, b₂ )^T   =   a₁b₁ + a₂b₂

b · a =  ( b₁, b₂ )^T · ( a₁, a₂ )^T   =   b₁a₁ + b₂a₂   =   a₁b₁ + a₂b₂

(Just in case your eyes have glazed over: notice that the stuff after the last "=" sign is the same in each case.)