Linear Algebra (Aljabar Linier) Week 1 Universitas Multimedia Nusantara Serpong, Tangerang Dr. Ananda Kusuma Ph: ,

Agenda Course Information Vectors

Course Overview Reference: –SAP (Satuan Acara Perkuliahan) Aljabar Linier UMN Assessment: –Quiz (minimal 2x)30% –Mid-semester Exam30% –Final Exam40%

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 Components of a vector

Vectors in the Plane

Vectors in R 3 : Although difficult to interpret geometrically, vectors exist in any n-dimensional space  R n is a set of all ordered n- tuples of real numbers written as row or column vectors. A vector v in R n is Vectors in R n : Definition or

Algebraic Properties of Vectors in R n

Vector Operation: Length Vector length (norm)  If we have a vector v=[v 1, v 2,..., v n ], then the length (or norm) of the vector is the nonnegative scalar defined by Example Let v=[2,1,6] in R 3

Vector Operation: Addition/Substraction If u=[u 1,u 2,...,u n ] and v=[v 1,v 2,...,v n ], then Example

Vector Operation: Scalar Multiplication Given a vector v=[v 1,v 2,...,v n ] and a real number c, then Example: If v=[-2,4], find 2v,1/2v,-2v

Vector Operation: Linear Combination A vector v is a linear combination of vectors v 1,v 2,...,v k, if there are scalars c 1,c 2,...,c k 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 (e 1 and e 2 ), or [-2,1] in uv system. Vector w is a linear combination of u and v (also e 1 and e 2 )

Vector Operation: Dot/Scalar/Inner Product If u=[u 1,u 2,...,u n ] and v=[v 1,v 2,...,v n ], then Properties Usage: length, distance, angle normalization can be conveniently described using the notion of the dot product Scalar

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 R n, Triangle Inequality  For all vectors u and v in R n,

Orthogonality Two vectors u and v in R n 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 R 3, u = [1,1,-2] and v = [3,1,2]. Are they orthogonal? Show that for all vectors u and v in R n, if u and v are orthogonal then

Cross Product The cross product is only valid in R 3, and it gives us a vector orthogonal to any two nonparallel vectors Show that e 1 x e 2 = e 3, e 2 x e 3 = e 1, e 3 x e 1 = e 2

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 R 3 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 R 3

Lines in R 3 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=[b 1,b 2,...,b n ] in The parity checked code vector is v=[b 1,b 2,...,b n,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.

The End Thank you for your attention!