A Note On Vectors...
A vector is an object in a multidimensional space. It is represented by its components measured
on a reference system. A reference system is a series of vectors from which the entire space can be
A commonly used mathematical notation for a vector is a lowercase bold letter – v, for
example. So, if the set of vectors u1,...,un is a reference system for a space with n dimensions, then
any vector of the space can be written as,
v = v1u1 + ··· + vnun,
where v1,…,vn are real numbers in the case of a real space. These numbers are called the components of a vector.
A matrix is a linear operator over vectors from one space to vectors in another space not necessarily of
the same dimension. This means that the application of a matrix on a vector is another vector. Matrix is
commonly represented with an uppercase bold letter – M, for example. The application of the matrix M on the
vector V is denoted by M • v. The fact that a matrix is a linear operator means that
M • (αu + βv) = αM • u + βM • v,
for any matrix M, any vectors u and v, and any numbers α and β.