FG GF x What we arrived at is that a linear transformation of a vector space may be expressed as a product of a matrix and a vector. Composition of two linear transformations is represented by a product of the corresponding matrices. The claim is more general than what was actually shown. R 2 is known as an arithmetic vector space. Horizontal tuples are usually called row vectors whereas the vertical ones are known as column vectors .
Matrices turn up as representing linear transformations for a fixed pair of bases.
To obtain the result of applying a transformation to a vector multiply the corresponding matrix and the tuple. Alexander Bogomolny 2412495 .