01-31-03 - Lecture 2 Andrei Antonenko January 31, 2003 1...

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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 + ··· + a n x n = b (1) Example 1.2. The equation 0 x 1 + 3 x 2 + 4 x 3 + 0 x 4 = 6 is a linear equation, but the equation 4 x 1 x 2 + 3 x 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, a i — they are called the coefficients of an equation; then, b — the constant of an equation; both a i ’s and b are given numbers. Also, we have x i ’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 ,...,k n ) such that a 1 k 1 + ··· + a n k n = b Example 1.4. Let’s consider the equation 0 x 1 + 3 x 2 + 4 x 3 = 7 1
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This note was uploaded on 05/28/2011 for the course AMS 2010 taught by Professor Andant during the Spring '03 term at SUNY Stony Brook.

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01-31-03 - Lecture 2 Andrei Antonenko January 31, 2003 1...

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