A plane is a two dimensional space and two vectors are sufficient to describe it, meaning that any point in th plane can be expressed as a sum of two vectors (relative to some fixed point). If therefore there are three vectors all of which lie in the plane, one of them is redundant and […]

## Eigenvalues and Eigenvectors

An eigenvectorof a matrixis such that if the vector is multiplied by the matrix, the result is a multiple of the vector. Eigenvectors are special and have many applications in may areas. We ,ay write In this equationis a constant called the eigenvalue. The procedure for finding eigenvectors and eigenvalues is quite simple. IfthenThis means […]

## Finding Factors in the Determinant of a Matrix

Often the determinant of a matrix may be written as a product of one or more factors. We can find an expression for the determinant andfactorise it, which may be long and tedious, or we may look at the matrix itself and see which changes we can make so that the matrix becomes linearly independent […]

## The Image of a Line Under the Transformation Represented by a Matrix

Any transformationrepresented by a matrixwhose entries are numbers is linear. The matrix will send a line to a line. The transformed line will in general be rotated and scaled, but it will still be a line. For example, suppose a transformationis represented by the matrix Consider the effect ofon the lineThe line may be represented […]

## Linear Independence and Volumes of Parallelpipeds

If a parallelogram is defined by two vectorsandthen the area of the parallelogram is defined by( Note whereis the angle betweenand If the vectorsandare joined by a third vectorto form a solid shape, then the volume of the solid is the area of the base (which we may consider to be the area of the […]

## FP4 – Direction Cosines

A vector may be described in a number of ways egand both describe the same vector. Both of the above notations are in a cartesian coordinate systems, but a vector may also be described in cylindrical and polar coordinates – both of this are more complicated. It is quite useful sometimes to be able to […]

## The Dot Product

The dot product of two vectorsandis a number. It is the length of the of vector projected onto the unit vector in the direction ofIfis the angle betweenand then from the diagram aboveThis equation is usually writtenorand this form can be used to find the angle between two vectors. To find the dot product of […]

## Group Generators

For each cyclic groupthere is an element – not necessarily unique – that generates the group, so that every element of the group is a result of repeated composition of some element with itself. If is such an element, and the grouphaselements, we can write A group does not have to be cyclic to have […]

## Subgroups

The Caylay table for the group of symmetries of a triangle, D-3 , is given below. The highlighted entries are special. They form a closed set, with each member of the set being composed with another member of the set to form some other member of the set. We say elements of a group which […]

## Lagrange’s Theorem

Ifis a finite subgroup with n elements, the number of elements in any subgroup ofmust divideThis is Lagrange‘s Theorem. Formally, Lagrange’s Theorem states, The order of any subgroup of a groupmust divide the order of Ifhas order 10, the only possible orders of any subgroup ofare 1, 2, 5 or 10. 1 is the order […]