Matrices/Linear system/Introduction/Section

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A system of linear equations can easily be written with a matrix. This allows us to make the manipulations which lead to the solution of such a system, without writing down the variables. Matrices are quite simple objects; however, they can represent quite different mathematical objects (e.g., a family of column vectors, a family of row vectors, a linear mapping, a table of physical interactions, a vector field, etc.), which one has to keep in mind in order to prevent wrong conclusions.


Let denote a field, and let and denote index sets. An -matrix is a mapping

If and , then we talk about an -matrix. In this case, the matrix is usually written as

We will usually restrict to this situation. For every ,  , , is called the -th row of the matrix, which is usually written as a row vector

For every ,  , , is called the -th column of the matrix, usually written as a column vector

The elements are called the entries of the matrix. For , the number is called the row index, and is called the column index of the entry. The position of the entry is where the -th row meets the -th column. A matrix with is called a square matrix. An -matrix is simply a column tuple (or column vector) of length , and an -matrix is simply a row tuple (or row vector) of length . The set of all matrices with rows and columns (and with entries in ) is denoted by , in case we also write .

Two matrices are added by adding entries with corresponding entries. The multiplication of a matrix with an element (a scalar) is also defined entrywise, so


The multiplication of matrices is defined in the following way.


Let denote a field, and let denote an -matrix and an -matrix over . Then the matrix product

is the -matrix, whose entries are given by

Such a matrix multiplication is only possible when the number of columns of the left-hand matrix equals the number of rows of the right-hand matrix. Just think of the scheme

the result is an -Matrix. In particular, one can multiply an -matrix with a column vector of length (the vector on the right), and the result is a column vector of length . The two matrices can also be multiplied with roles interchanged,


An -matrix of the form

is called a diagonal matrix.


The -matrix

is called identity matrix.

The identity matrix has the property , for an arbitrary -matrix .


If we multiply an -matrix with an column vector , then we get

Hence, an inhomogeneous system of linear equations with disturbance vector , can be written briefly as

Then, the manipulations on the equations, which do not change the solution set, can be replaced by corresponding manipulations on the rows of the matrix. It is not necessary to write down the variables.