If I want to multiply two vectors by means of matrices, one of them must be transposed in order to perform multiplication. Does it matter which vector should be transposed or can I choose randomly (that sounds incorrect) which vector to be transposed?
Best regards, Sergey
You say you want to "multiply vectors". Which multiplication? There are three different kinds of multiplications used with vectors- the "dot product" (also called "inner product"), "cross product" and "scalar product". Since you say "vectors", plural, and the scalar product involves one vector and one scalar, I will drop the scalar product. The cross product is "anti-commutative", uxv= -vxu so certainly changing the order will change that. For the dot product we can do either $\begin{pmatrix}a & b & c\end{pmatrix}\begin{pmatrix}p \\ q \\ r\end{pmatrix}= ap+ bq+ cr$ or $\begin{pmatrix}p & q & r\end{pmatrix}\begin{pmatrix}a \\ b \\ c \end{pmatrix}= pa+ qb+ rc$, the same thing!
But $\begin{pmatrix}a \\ b \\ c \end{pmatrix}\begin{pmatrix}p & q & r\end{pmatrix}= \begin{pmatrix}ap & aq & ar \\ bp & bq & br \\ cp & cq & cr \end{pmatrix}$ is completely different!