Basics of Linear Algebra

A review?

Matrix

Mathematical term essentially corresponding to an array An arrangement of numbers into rows and

columns. Each row is the same length Each column is the same length Usually, we specify a position in the matrix as

row, column

Vector

A one dimensional matrix One row (or one column)

We can treat it as a special case of a matrix

Matrix operations

Matrix (vector) addition/subtraction Add/subtract the corresponding elements in two

matrices (vectors) of exactly the same size and shape. If A = and B =

What is A + B? Matrices can only be added or subtracted if they are

exactly the same size and shape.

103 120 115

210 230 204

301 300 320

15 17 45

32 44 47

62 25 17

Multiplication

If A is an m x n matrix and B is an n x p matrix, then the product AB is an m x p matrix whose elements are defined by cij = ∑aikbkj

That is, sum the term by term products of the elements in row I of A column j of B.

k = 1

n

Transposition

If A is m x n, the transpose of A is n x m. The rows become columns and the columns

become rows. This is sometimes needed to put things into a

form that is compatible for multiplication.

Properties

The identity matrix has 1s on the main diagonal and 0s elsewhere. Multiplication by the identity matrix yields the original

matrix. i.e. AI = IA = A The size of the identity matrix is made to be compatible

for the operation intended. The zero matrix has 0 in every position. If A, B, C are of appropriate sizes, then

A(BC) = (AB)C A(B+C) = AB +AC (A+B)C = AC + BC

Matrix inverse

The inverse (A-1) is defined such at A A-1 is I. Not every matrix has an inverse. If no inverse exists,

then the matrix is called singular (non invertible) If A is nonsingular, so is A-1

If A, B are nonsingular, then AB is also non singular and (AB) -1 = B -1A -1 (Note reversed order.)

If A is nonsingular, then so is its transpose and (AT ) -1 = (A-1)T

Vectors and Vector Spaces A vector with 2 elements (a 2-vector) is written as( ). The vector is represented by a line in a plane, starting

at the origin and ending at the point x,y. For x=2 and y=1:

The length of the vector is calculated using the Pythagorean theorem: ||vector|| = √ x2 + y2

xy

(2,1)

Vector operations

Addition and subtraction consist of adding or subtracting the corresponding elements. Only vectors of the same size can be added/subtracted.

What does the sum of two vectors look like in the coordinate system?

Vector angles

There is an angle between any two vectors in the coordinate system. One way of comparing the vectors is to measure the angle between them. Cos =

Where ||X|| is the length, or magnitude, of X

X = ( ) Y = ( )

x1x2 + y1y2

||X|| ||Y||

x1

y1

x2

y2

Dot product and cosine

The dot product of vectors X, Y is defined as X * Y = x1x2 + y1y2

So, Cos =

If the cosine is 0, the vectors are at right angles.

X * Y

||X|| ||Y||

Larger dimensions

It is easy to visualize vectors in two space, but larger dimensions are also useful. We cannot draw them so easily, but the properties of length, distance, etc. remain interesting.

We can draw 3-vectors. Note that we cannot express the x axis in terms of the y

axis. In 3-space, we cannot express any of x, y, z in terms of just the others. The three axes define the space.

When we look at the relationships between vectors that define index terms or queries, the relationship between them tells us whether they represent totally unrelated information, or information that is more or less related.

Reference for this review: Introductory Linear Algebra with Applications

Bernard Kolman Macmillan Publishing, 1984 Chapters 1 and 3.