MAST10007 Linear Algebra
THE UNIVERSITY OF MELBOURNE
SCHOOL OF MATHEMATICS AND STATISTICS
Summer 2022 Lecture Notes
These notes have been made in accordance with the provisions of Part VB of the copyright act for teaching purposes of the
University. These notes are for the use of students of the University of Melbourne enrolled in MAST10007 Linear Algebra.
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Topic 1: Linear equations [AR 1.1 and 1.2]
1.1. Systems of linear equations. Row operations.
1.2. Reduction of systems to row-echelon form.
1.3. Reduction of systems to reduced row-echelon form.
1.4. Consistent and inconsistent systems.
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1.1 Systems of linear equations. Row operations
Linear equations
Definition (Linear equation and linear system)
A linear equation in n variables, x1, x2, . . . , xn, is an equation of the form
a1x1 + a2x2 + · · ·+ anxn = b,
where a1, . . . , an a
nd b are constants.
A finite collection of linear equations in the variables x1, x2, . . . , xn is
called a system of linear equations or a linear system.
Example
x1 + 5×2 + 6×3 = 100
x2 x3 = 1
x1 + x3 = 11
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Definition (Homogeneous linear system)
A system of linear equations is called homogeneous if all the constants
on the right hand side are zero.
Example
x1 + 5×2 + 6×3 = 0
x2 x3 = 0
x1 + x3 = 0
Definition (Solution of a system of linear equations)
A solution to a system of linear equations in the variables x1, . . . , xn is a
set of values of these variables which satisfy every equation in the
system.
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Example 1. Data fitting using a polynomial
Find the coefficients c , α, β, γ such that the cubic polynomial
y = c + αx + βx2 + γx3 passes through the points specified below.
x y
0.1 0.90483
0 1
0.1 1.10517
0.2 1.2214