Transcript Linear Algebra

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.
15 17 45
 If A = 103 120 115 and B =
210
230
204
32 44 47
301
300
320
62 25 17
 What is A + B?
 Matrices can only be added or subtracted if they are
exactly the same size and shape.
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
n
 cij = ∑aikbkj
k=1
 That is, sum the term by term products of the
elements in row I of A column j of B.
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
x
as
y
 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:
( ).
(2,1)
 The length of the vector is calculated using the
Pythagorean theorem: ||vector|| = √ x2 + y2
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  = x x + y y
1 2
1 2
||X|| ||Y||
Where ||X|| is the length, or magnitude, of X
 X=
x1
(y )
1
Y=(
x2
y2
)
Dot product and cosine
 The dot product of vectors X, Y is defined as
 X * Y = x1x2 + y1y2
 So, Cos  =
X*Y
||X|| ||Y||
 If the cosine  is 0, the vectors are at right angles.
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.