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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