MATH 221

Linear Equation

A linear equation is any equation that takes the form

Where

are variables
are coefficients
is a constant

Examples

Non-Examples

Linear System

a system of linear equations (or a linear system) is a collection of one or more linear equations involving the same variables.

An example of a linear system is the following:

The coefficients of the left side can be written as

The other side of the equation can be represented using the column vector

The variables can then be represented using a column vector as well

This means the linear system can also be expressed using matrices as follows

In this example the matrices are being multiplied using the dot product

Usually this linear system is just written simply as

Consistency

Linear systems which have no-solutions are called inconsistent, conversely they're called consistent if they have at least one solution.

Solving

Solving linear systems can be done in a variety of ways

Gaussian Elimination

Vector

Vectors are any quantity that can not be represented by a single number. Vectors are typically represented by an arrow pointing in the direction of the vector with the length of the arrow representing the magnitude of the vector.

Example Vector

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Vectors can be written using different notations and have a set of intrinsic properties, these vectors can also be transformed using different operations.

Vector Addition

Vector addition is the process of combining two or more vectors to produce a resultant vector.

Vector addition can be represented geometrically in 2-d as follows:

Geometric Vector Addition

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Notice that vector addition is commutative as which is also true of scalaraddition.

Vector addition can also be defined using matrix notation as follows:

Vector Subtraction

Vector subtraction can be performed by taking the inverse of the second vector and adding it the first vector. This is defined formally as

Vector addition can also be defined using matrix notation as follows:

Scalar Multiplication

A vector can be multiplied (sometimes also called stretched or scaled) by a scalar, this has the effect of multiplying all the components of the vector by the scalar.

This operation can be defined using matrix notation as follows:

where is the scalar is being multiplied by.

Multiplying a vector by has the effect of flipping the vector around.

Linear Combination

A linear combination is the expression constructed from a set of constituting vectors where each vector is multiplied by some scalar and then summed together.

Formally given a set of the linear combination is any equation of the form

where is some scalar constant.

Two vectors are said to be "linear combinations" of each other if there exists some integer such that .

Collinearity

Collinearity refers to objects which are said to "lie on a single line". Specifically vectors are collinear if they are linear combinations of each other, the collinearity of points are also defined in the same way.

Set Builder Notation

Set builder notation is a notation for describing a set by stating the properties that it's elements must satisfy.

The set builder notation has three parts, a variable a vertical bar separator "", and a predicate, all three of these parts are contained in curly brackets. For example:

Additionally, the domain of can be explicitly stated on either the left or right side of the separator:

Domain expressed on the left side of the separator

Domain expressed on the right side of the separator, using a logical conjunction with the predicate.

In both of these examples represents set membership.

Example

The set of all even integers can be defined as follows using set builder notation and the existential quantifier:

Linear Span

The linear span of a set of vectors is all of it's constituting linear combinations. The span of vectors is typically denoted as . Given a set of vectors, their span is defined as follows

Determining if a Vector is in the Span of Others

Given a vector we can determine if it is in the span of , by checking if the following equation has any non trivial solutions

Example

Given the following set of vectors

determine if the span of them contains .

First we setup the equation

Then the equation is converted into the standard linear system form

we can then perform gaussian elimination to get our final solutions

Linear Independence

A set of vectors is said to be linearly independent if there exists no vector in the set that is a linear combination of any other vector in the set. If such a vector exists the set of vectors is said to be linearly dependent.

Formally a set of vectors is linearly independent if the vector equation

has only the trivial solution . The set is linearly dependent otherwise (ie. if there is a solution where any of is nonzero).

Quick Checks

Dimensionality

If the length of the set is greater than the dimensionality of the vectors within it, the set is automatically linearly dependent.

Zero Vectors

If any of the vectors in the set are , the whole set must be linearly dependent.

Determinant

If the length of the set is equal to the dimensionality of the vectors we can check the determinant of the vectors arranged in a matrix to determine linear independence. If the determinant is nonzero they are independent.

Eigenvector

An eigenvector is a nonzero vector who's direction is unchanged (or reversed) by a linear map. The quantity that an eigenvector is scaled by under the mapping is called the eigenvalue and is written as a . Eigenvectors only exist for square matrices as for others they would either be all vectors or .

Formally given a linear map which is a square matrix the vector and the scalar are an eigenvector and eigenvalue of if the following equation holds true:

Eigenspace

An eigenspace of a linear map is the set of all eigenvectors of the map for a given eigenvalue and the zero vector.

Characteristic Polynomial

The characteristic polynomial of a square matrix is the polynomial formed with roots at the eigenvalues of the matrix . The characteristic polynomial of square matrix is defined as

which expands out to an degree polynomial in . The roots of (where ) are the eigenvalues of .

Factoring the Characteristic Polynomial

Special Cases

Matrix

For a matrix it's characteristic polynomial can be calculated as follows:

Triangle Matrix

For triangular matrices of size their characteristic polynomials can be calculated as follows:

This means that for triangular matrices their eigenvalues are just their diagonal entries.

Example

Find the characteristic polynomial of the matrix

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