Download Sec 1.1 - UBC Math

Math 221, Section 1.1
Today: (1) setting up terminologies, notations, ideas, ...
(2) solve a system of linear equations
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Linear equations
A linear equation in n variables/unknowns x1 , . . . , xn is an
equation that can be written in the form
a1 x1 + · · · + an xn = b
where the coefficients a1 , . . . , an and the constant term b are
known in advance.
Examples
equation in 2 variables ax + by = c
equation in 3 variables ax + by + cz = d.
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System of linear equations
A system of linear equations (or a linear system) is a collection
of one or more linear equations.
Examples in the book
I
2x − y + 1.5z = 8
x
− 4z = −7
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x − 2y + z = 0
2y − 8z = 8
−4x + 5y + 9z = 9
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Solutions
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A solution of the system is an array (s1 , . . . , sn ) of numbers
that makes each equation a true statement when the values
(s1 , . . . , sn ) are substituted for the variables (x1 , . . . , xn )
respectively.
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The set of all possible solutions is called the solution set of
the linear system.
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3 situations happen: exactly one/unique solution, no solution,
or infinitely many solutions
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Solutions
An example in the book
2x − y + 1.5z = 8
x
− 4z = −7
I
has a solution (−7, −22, 0),
I
but the solution set is (−7 + 4t, −22 + 9.5t, t), where t is any
real number.
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Matrix Notation
Write the system
x − 2y + z = 0
2y − 8z = 8
−4x + 5y + 9z = 9
as an augmented matrix


1 −2 1 0
 0
2 −8 8 
−4 5
9 9
[coefficient matrix | constant vector]
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Solving a Linear System
Solving the system by elimination
x − 2y + z = 0
2y − 8z = 8
−4x + 5y + 9z = 9
VS
Reducing the matrix by row operations


1 −2 1 0
 0
2 −8 8 
−4 5
9 9
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Elementary row operations
What kind of operations did we apply to a matrix?
3 elementary row operations
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(Replacement) Add to one row a multiple of another row.
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(Scaling) Multiply all entries in a row by a nonzero constant.
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(Interchanging) Interchange two rows.
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Consistency
A system of linear equations is said to be consistent/solvable if it
has either one solution or infinitely many solutions.
A system is inconsistent/unsolvable if it has no solution.
Examples
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Consistency
Examples
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