Linear Algebra(Aljabar Linier)

Week 1

Universitas Multimedia NusantaraSerpong, Tangerang

Dr. Ananda Kusumae-mail: ananda_kusuma@yahoo.com

Ph: 081338227031, 081908058069

Course Overview

• Reference:

– SAP (Satuan Acara Perkuliahan) Aljabar Linier UMN

• Assessment:

– Quiz (minimal 2x) 30%– Mid-semester Exam 30%– Final Exam 40%

Textbook

David Poole, Linear Algebra: A Modern Introduction, Thomson Brookscole (second edition), 2006

Why Should You Study Linear Algebra

• Linear algebra looks very abstract , but it has many real-life applications; e.g. for computer scientists/engineers:

– Networks:• Circuit theory, telecommunication network, transportation network

– Coding Theory– Graph Theory– Computer Graphics– Image Compression– Optimization: Linear Programming– Searching the Internet

Vectors

• A vector is a quantity that has both a magnitude and a direction Vectors are equal if they have the same magnitude/length and direction

• Example:

• Column notation:

Vector Notation

4

2v

Components of a vector

Vectors in the Plane

?a?b

?c

2),25( v

• Vectors in R3:

• Although difficult to interpret geometrically, vectors exist in any n-dimensional space Rn is a set of all ordered n-tuples of real numbers written as row or column vectors.

• A vector v in Rn is

Vectors in Rn : Definition

nvvv ,...,, 21

nv

v

v

2

1

or

Algebraic Properties of Vectors in Rn

Vector Operation: Length

• Vector length (norm) If we have a vector v=[v1, v2, ..., vn], then the length (or

norm) of the vector is the nonnegative scalar defined by

• Example• Let v=[2,1,6] in R3

222

21 nvvvv

v

41v

Vector Operation: Addition/Substraction

• If u=[u1,u2,...,un] and v=[v1,v2,...,vn], then

• Example

nn vuvuvuvu ,...,, 2211

Vector Operation: Scalar Multiplication

• Given a vector v=[v1,v2,...,vn] and a real number c, then

• Example: If v=[-2,4], find 2v,1/2v,-2v

nn vvvvvvv c,c,c,...,,cc 2121

Vector Operation: Linear Combination

• A vector v is a linear combination of vectors v1,v2,...,vk, if there are scalars c1,c2,...,ck such that

• Example: Coordinate System. Let u = [2,1] , v = [1,3].• Define new cooordinate system based on u and v. • w=-2u+v=-2[2,1]+[1,3]=[-3,1]• Vector w=[-2,1] in standard coordinate system (e1 and e2), or [-2,1] in

uv system. Vector w is a linear combination of u and v (also e1 and e2)

kkvcvcvcv ...2211

Vector Operation: Dot/Scalar/Inner Product

• If u=[u1,u2,...,un] and v=[v1,v2,...,vn], then

• Properties

• Usage: length, distance, angle normalization can be conveniently described using the notion of the dot product

nnvuvuvuvu ...2211 Scalar

vvv vuvud ),( vu

vu cos

Cauchy-Schwarz Inequality and Triangle Inequality

The dot product allows us to derive two important mathematical inequalities:

• Cauchy-Schwarz Inequality: For all vectors u and v in Rn,

• Triangle Inequality For all vectors u and v in Rn,

vuvu

vuvu

Orthogonality

• Two vectors u and v in Rn are orthogonal to each other if

• In R2 or R3, two nonzero vectors u and v are perpendicular if the angle between them is a right angle, i.e.

• In R3, u = [1,1,-2] and v = [3,1,2]. Are they orthogonal?

• Show that for all vectors u and v in Rn, if u and v are orthogonal then

0 vu

090cos 0 vu

vu0 vu

222vuvu

Cross Product

• The cross product is only valid in R3, and it gives us a vector orthogonal to any two nonparallel vectors

• Show that e1 x e2 = e3, e2 x e3 = e1, e3 x e1 = e2

Vectors and Geometry

Lines in 2 Dimensions

• Use our knowledge of vectors to describe lines and planes

Definition

Example: Line 2x + y = 0

What is the normal norm of the equation of a line ?

Example: Line 2x + y = 5

What is the normal norm of the equation of a line ?

Example: Line 2x + y = 0

What is the vector form of the equation of a line ?

Planes in R3

Find the normal form of a plane that contains the point P=(6,0,1) and has normal vector n = [1,2,3].

Planes in R3

Lines in R3

The intersection of two planes is a line

Code Vectors

• Modulo 2 arithmetic:

• Binary Code:

Error Detecting Codes

• E.g how to detect a single error:• Suppose the message is the binary vector b=[b1,b2,...,bn] in• The parity checked code vector is v=[b1,b2,...,bn,d] in , where the

check digit (parity) d is chosen so that

• Remember dot/inner product? The dot product of vector 1 = [1,1,...,1] and the received vector v. If the dot product equal to 1, then there is a single error.

n2

12n

011 dbbb n

The End

Thank you for your attention!