A system of linear equations is a collection of m equations in the variable quantities [image] of the form

[image]

[image]

[image]

[image]

where [image]

A solution of a system of linear equations of n variables [image] is an ordered list of n complex numbers, [image], such that if we substitute [image] then for every equation of the system, the left side will equal the right side i.e. each equation is true simultaneously.

A solution set of a system of linear equations is the set that contains every solution to the system. A solution set can be empty or infinite.

Two systems of linear equations are equivalent if their solutions sets are equal.

Given a system of linear equations, the following three operations will transform the system into a different one, and each operation is known as an equation operation.

1. Swap the locations of two equations in the list of equations.
2. Multiply each term of an equation by a non-zero quantity.
3. Multiply each term of one equation by some quantity, add the result to a second equation.

An m x n matrix is a rectangular layout of numbers from C having rows m and columns n.

For a matrix A, the notation [image] (or [image]) refers to the complex number in row i, column j of A.

A column vector (often just called a vector) of size m is an ordered list of m numbers written vertically from top to bottom. Denoted as [image], etc. somestimes as [image].

An entry in a vector is denoted as [image]

[image]

A column vector whose entries are all zero.

[image]

For a system of linear equations the cofficient matrix is the m x n matrix

[image]

For a system of linear equations, the vector of constants is the column vector of size m

[image]

For a system of linear equations, the solution vector is the column vector of size n. (It can also double as a solution set).

[image]

If A is a coefficient matrix of a system of linear equations and b is the vector of constants, then LS(A,b) is the matrix representation of a linear system.

Suppose we have a system of m equations in n variables with coefficient matrix A and vector of constants b. Then the augmented matrix is the m x (n + 1) matrix whose first n columns are the columns of A and whose last column (n + 1) is the column vector b. This matrix will be written as (A|b).

[1 1 1 1 | 6]
[1 -4 0 0 | 0]
[4 4 2 2 | 2]

There are three row operations (similar to Equation Operations).

[image]   swap rows

[image]       multiply row by a scalar

[image]   multiply row i by a scalar and add the

           result to row j

Two matrices A and B are row equivalent if one can be obtained from the other by a sequence of row operations.

A matrix is in reduced row echelon form if it meest the following conditons:

1. Any all zero rows are listed below all non-zero rows.
2. The left most non-zero entry of a row is equal to 1
3. The left most non-zero entry of a row is the only non-zero entry in its column.
4. For any two different left most non-zero entries, one in row i column j, the other in row s, column t. If s > i, then t > j.

A column containing a leading 1 is called a pivot column. The number of non-zero rows  is represented by r and is also the number of pivot columns.

The set of column indices for the pivot columns will be denoted by [image] where [image] while the columns that are not pivots will be denoted by [image] where [image].

Example:

[image]

There are 3 pivot columns: 1, 2 and 4.

There are 2 free columns: 3 and 5.

A system of linear equations is consistent if it has at least one solution. Otherwise, the system is called inconsistent.

Suppose A is the augmented matrix of a consistent system of linear equations and B is a row-equivalent matrix in row-reduced echelon form. Suppose j is the index of a pivot column of B. Then the variable x_j is dependent. A variable that is not dependent is called independent or free.

A system of  linear equations, LS(A,b), is homogeneous if the vector of constants is the zero-vector, in other words, if b=0.

[homogeneous: composed of parts or elements that are all of the same kind]

Suppose a homogeneous system of linear equations has n variables.

The solution [image] (i.e. x = 0) is called the trivial solution.

The null space of a matrix A, denoted by N(A), is the set of all the vectors that are solutions to the homogeneous system LS(A,0).

A matrix with m rows and n columns is square if m=n. In this case, we say the matrix has size n. To emphasize a situation when the matrix is not square we will call it rectangular.

Suppose A is a square matrix. Supose further that the solution set to the homogeneous linear system of equations LS(A,b) is {0}, in other words, the system has a trivial solution. Then we say tha A is a nonsingular matrix. Otherwise, we say A is a singular matrix.

[The terms singular and nonsingular only apply to square matrices and only to matrices, not systems of linear equations.]

The m x n identity matrix, [image], is defined by

[image]

The vector space [image] is the set of all column vectors of size m with entries from the set of complex numbers.

When a set similar to this is defined using only column vectors where all the entries are from the real numbers it is written [image] and is known as Euclidean m-space.

Suppose that [image]. Then [image] and [image] are equal, written [image], if

[image]

Suppose that [image]. The sum of [image] and [image] is the vector [image] defined by

[image]

Suppose [image] and [image]. Then the scalar multiple of [image] by [image] is the vector [image] defined by

[image]

Given n vectors [image] and n scalars [image], their linear combination is the vector [image].

Given a set of vectors [image], their span, (S), is the set of all possible linear combinations of [image]. Symbolically,

[image]

[A span is a construction that begins with a finite collection of vectors S (of size p) from which is built the description of an infinite set (S)]

Given a set of vectors [image] a true statement of the form

[image]

is a relation of linear dependence on S. If this statement is formed in a trivial fashion i.e.

[image]

then we say it is the trivial relation of linear dependence.

[NOTE that the relation of linear dependence is an equation though most of it is a linear combination.]

The set of vectors [image] is linearly independent if there is a relation of linear dependence on S that is not trivial. In the case where the only relation of linear dependence on S is the trivial one, the S is a linearly independent set of vectors.

Linear independence is a property of a set of vectors.

Suppose that u is a vector from [image]. Then the conjugate of the vector, [image], is defined by

[image]

Given the vectors [image] the inner product of  u and v is the scalar quantity in [image]

[image]

Note: the notation for the inner product operation, [image], uses the same left and right angle brackets used for a spanning set, [image]

The norm of a vector is the scalar quantity in [image]

[image]

A pair of vectors [image] from [image] are orthogonal if their inner product is zero, [image]

Suppose that [image] is a set of vectors from [image]. Then S is an orthogonal set if every pair of different vectors from S is orthogonal, that is [image]

Let [image] denote the column vectors defined by

[image]

Then the set [image] is the set of standard unit vectors.

[This is essentially the definition of an Identity Matrix, [image]. [image] is the index to a pivot column in an m x n matrix.]

Suppose [image] is an orthogonal set of vectors such that [image] for all [image]. Then S is an orthonormal set.

[Multiply each vector in the orthogonal set by the reciprocal of its norm, [image], and the resulting vector will have a norm of 1. The scaling does not effect the orthogonal properties of the original set.]