MATH136 Lecture Notes - Lecture 9: Coefficient Matrix, Augmented Matrix, Oberheim Matrix Synthesizers
14 May 2018
School
Department
Course
Professor
Friday, May 19
−
Lecture 9 : Systems of linear equations. (Refers to 2.1)
Concepts:
1. m linear equations in n unknowns (system of linear equations).
2. solving a system of linear equations
3. equivalent system of linear equations
4. coefficient matrix, augmented coefficient matrix.
5. RREF coefficient matrix, pivot, leading-1, REF matrix
6. Rank, vector eqn of a solution, basic unknowns, free unknowns, consistent
9.1 Introduction − Linear equations are equations of the form
a1x1 + a2x2 + ... + anxn = b.
where the xi’s are called the variables, or unknowns, while the ai’s are the coefficients of
the linear equation. The term b which is not juxtaposed to a variable is referred to as the
constant. Note that the ai’s and b can possibly be zeroes. In the case 0x1 + 0x2 + ... + 0xn =
0, we will refer to this linear equation as the “trivial linear equation”. Every point
(x1, x2, ..., xn) in ℝn satisfies it. Such an equation can never be a plane or hyperplane
(This is not mentioned in the text.)
A system of linear equations is a finite set of linear equations each of which contains
the same variables.
“Solving a system of m linear equations in n variables” means finding the set of all
vector values x = (x1, x2, x3, …, xn) which simultaneously satisfy all of the m linear
equations.
In this lecture we discuss algorithms for solving these by manipulating only the
coefficients and the constants.
To solve a system of linear equations we apply repeatedly the following fundamental
mathematical principles:
9.1.1 Example − In the following example we solve a system of three linear equations
in three unknowns by applying these principles.
Solve the system
4x − 8y + −4z = 36
2x − y + z = 6
3x − 2y + −2z = 2.
Solution : Denote each equation as follows :
Each of the systems listed above have the same solution. The last system is the one
that is easiest to solve.
9.1.2 Definition – Two systems of linear equations are said to be equivalent systems
if they have the same non-empty set of solutions.
In the example of above we have step by step transformed a system of equations to
another. Each time we this transformation did not change the solution set of the
system. Each one of the systems of equations which appear above are equivalent
systems.
9.1.3 Geometric interpretation.
An equation with three unknowns describes a plane in 3-space.
When three equations in three unknowns are considered simultaneously to form a
“system of equations” the solution set to this system describes all points which
belong to the three planes. This solution set can be empty, a single point, a line or
a plane.
Representing a system of linear equations as a rectangular array of numbers.
9.2 Definition − System of linear equations. Consider the following general system of m
linear equations with n unknowns:
a11x1 + a12x2 + ... + a1nxn = b1.
a21x1 + a22x2 + ... + a2nxn = b2.
...
am1x1 + am2x2 + ... + amnxn = bm.
where b1, b2, ..., bm, are the constants and x1, x2, ..., xn are unknowns. The aij's, 1 ≤ i ≤ m
and 1 ≤ j ≤ n, are numbers which we call coefficients.
The array of coefficients of the system expressed as follows
often written as
A = [aij]m × n
is called the coefficient matrix A of the given system. If we add an extra column
containing the bi's to the right of the matrix A we obtain a larger matrix
34
MATH136 Full Course Notes
Verified Note
34 documents
Document Summary
Friday, may 19 lecture 9 : systems of linear equations. (refers to 2. 1) 9. 1 introduction linear equations are equations of the form a1x1 + a2x2 + + anxn = b. where the xi"s are called the variables, or unknowns, while the ai"s are the coefficients of the linear equation. The term b which is not juxtaposed to a variable is referred to as the constant. Note that the ai"s and b can possibly be zeroes. In the case 0x1 + 0x2 + + 0xn = 0, we will refer to this linear equation as the trivial linear equation . Every point (x1, x2, , xn) in n satisfies it. Such an equation can never be a plane or hyperplane (this is not mentioned in the text. ) A system of linear equations is a finite set of linear equations each of which contains the same variables.
