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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 generated.

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.

and Matrices...

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.

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