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Lecture 2
Andrei Antonenko
January 31, 2003
1 Introduction to linear equations
Last lecture we were talking about the general mathematical concepts, like a concept of a number, a concept of a set, a concept of an operation. This lecture we will start studying concepts of linear algebra itself. The first main thing which appears in linear algebra is a linear equation.
Definition 1.1. The linear equation is the equation which can be transposed to the form
a 1 x 1 + a 2 x 2 + · · · + anxn = b (1)
Example 1.2. The equation 0 x 1 + 3x 2 + 4x 3 + 0x 4 = 6
is a linear equation, but the equation
4 x 1 x 2 + 3x 3 = 5
is not, since it contains the term 4 x 1 x 2 , and so can not be transposed to the form (1).
Here, we have 3 different types of numbers. First, ai — they are called the coefficients of an equation; then, b — the constant of an equation; both ai’s and b are given numbers. Also, we have xi’s — variables, which are not given, but should be determined from this equation. Our main goal is to solve this equation. What does it mean “to solve” the equation???
Definition 1.3. A solution of a linear equation (1) is a n-tuple of numbers (k 1 , k 2 ,... , kn) such that a 1 k 1 + · · · + ankn = b
Example 1.4. Let’s consider the equation
0 x 1 + 3x 2 + 4x 3 = 7
Then (1, 2 , 3) is not a solution for this equation since 0 · 1 + 3 · 2 + 4 · 3 = 18 6 = 7, and (1, 1 , 1) is a solution, since 0 · 1 + 3 · 1 + 4 · 1 = 7. Moreover, ∀k ∈ R, (k, 1 , 1) is a solution for this equation. So, we see, that this equation has infinitely many solutions.
“To solve” the equation is to find all its solutions.
2 The easiest equation
Now we’ll proceed to the easiest type of linear equation — linear equation with one variable. It is an equation of the following form: ax = b (2) Let’s analyze it. We have 3 cases for this equation.
- Case 1. a 6 = 0, b 6 = 0. In this case the equation has only one solution, which can be obtained by the following formula x = ba This is a solution. To check it let’s substitute this value to the equation; we will have: a · (^) ab = b. Moreover this is a unique solution: multiplying ax = b by (^1) a we get x = b · (^1) a = ba.
- Case 2. a = 0, b = 0. In this case left hand side of the equation is equal to 0 for any x, and right hand side is 0. So, any number is a solution of this equation. So, in this case the equation has infinitely many solutions.
- Case 3. a = 0, b 6 = 0. In this case left hand side of the equation is equal to 0 for any number x, and right hand side is not equal to 0. So, in this case the equation has no solutions.
So, we saw 3 different cases — the equation could have 0, 1 or infinitely many solutions. Later we will see that this is the general case.
3 General linear equation
Now we will proceed to the general form of a linear equation. As we’ve already seen, it can be written in the form a 1 x 1 + a 2 x 2 + · · · + anxn = b. (3)
We’ll get the general solution for this equation. First we’ll need some definitions.
and any n-tuple (k 1 ,... , kn) is a solution. Case 2. b 6 = 0. Then the equation has the form
0 x 1 + · · · + 0xn = b,
and the left hand side of an equation is always 0, but right hand side is not. So the equation has no solution. As we can see, we got the same result, as in the easiest case: The equation can have 0, 1 or infinitely many solutions.
