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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
• 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