1. MATH 1046 Introduction to Linear Algebra
Alex Karassev

2. 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 cx in S
A function L from S to real numbers is called linear if L(x+y) = L(x) + L(y) and L(cx) = cL(x) for all x,y and c

3. 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

4. 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)

5. 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

6. 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)

7. 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

8. 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

9. Modern Applications Economics (linear optimization problems)
Computer graphics
Search engines