* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download MATH 1046 Introduction to Linear Algebra
Survey
Document related concepts
Transcript
MATH 1046 Introduction to Linear Algebra Alex Karassev What is “linear”? • Set S such that • We can add elements of S, i.e. for x and y in S there exists x+y in S • We can multiply element of S by real numbers, i.e. for any x in S and any real number c there exists c x in S • A function L from S to real numbers is called linear if L(x+y) = L(x) + L(y) and L(c x) = cL(x) for all x,y and c Example • Let S be the set of all differentiable functions defined on real numbers • For any function f from S, let L(f) = f’(0) • Then L is linear Linear algebra studies • Sets S with addition and multiplication by numbers (and some additional properties), called vector spaces (or linear spaces) • Linear functions and their generalizations (called linear operators) Vectors on the plane • A vector on the plane is defined by two points, A and B, and their order • Notation: AB • Two vectors are considered equal if they – are parallel – have the same length – have the same direction Vectors on the plane • Vectors on the plane can be added (using parallelogram rule) and multiplied by constants • Although the idea of the parallelogram rule appeared already in antiquity, and implicitly the idea of vector quantities was used in physics for a few centuries, the systematic study of vectors did not begin until 19th century (in connection with complex numbers) In this course, main objects of study will be: • Vectors, represented as ordered sequence's of numbers (x1,x2,…,xn) • Matrices – rectangular tables of numbers Traditional Applications • Systems of linear equations • Use of vectors in physics to represent velocity, acceleration, forces, etc. • Linearization in calculus: local approximation of a function by a linear function Modern Applications • Economics (linear optimization problems) • Computer graphics • Search engines