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# System of Linear Equations

Slide 1

 Definition 1 Fix a set of numbers , where i = 1, ...,m and j = 1,... , n. A system of m linear equations in n variables , is given by Consistent: It has solutions (one or infinitely many). Inconsistent: It has no solutions.

Slide 2

 Solutions to a system of linear equations can be obtained by: • Substitution. (Convenient for small systems.) • Elementary Row Operations . (Convenient for large systems.) It is not needed to write down the variable while performing the EROs. Only the coefficients of the system , and the right hand side is needed. This is the reason to introduce the matrix notation.

Slide 3

Matrix Notation

 System of Equations: Augmented Matrix Slide 4

 Elementary Row Operations (EROs) • Add to one row a multiple of the other. • Interchange two rows . • Multiply a row by a nonzero constant. EROs do not change the solutions of a linear system of equations. EROs are performed until the matrix is in echelon form. Echelon form: Solutions of the linear system can be easily read out.

Slide 5

 Definition 2 The diagonal elements of a matrix are given by , for i from 1 to the minimum of m and n. Examples: Only the diagonal elements are given Slide 6

 Echelon Forms • Echelon form: Upper triangular. (Every element below the diagonal is zero .) • Reduced Echelon Form: A matrix in echelon form such that the first nonzero element in every row satisfies both, - it is equal to 1, - it is the only nonzero element in that column.

Slide 7

 Existence and uniqueness • A system of linear equations is inconsistent if and only if the echelon form of the augmented matrix has a row of the form • A consistent system of linear equations contains either, - a unique solution, that is, no free variables, - or infinitely many solutions, that is, at least one free variable.

Slide 8

 Vectors in IRn • Definition, Operations, Components. • Linear combinations. • Span.

Slide 9

 Definition, Operations, Components Definition 3 A vector in IRn, n≥1, is an oriented segment. Operations: • Addition, Difference : Parallelogram law . • Multiplication by a number: Stretching, compressing. In components: Slide 10

 Some properties of the addition and multiplication by a scalar: u + v = v + u, u + (v + w) = (u + v) + w, a(u + v) = au + av, (a + b)u = au + bu.

Slide 11

 Definition 4 A vector w ∈ IRn is a linear combination of p≥ 1 vectors in IRn if there exist p numbers such that A system of linear equations can be written as a vector equation: Slide 12

 Span Definition 5 Given p vectors in IRn, denote by Span the set of all linear combinations of . Note: • Span , • If w ∈ Span , then there exist numbers such that Slide 13

 Matrices as linear functions Definition 6 A linear function y : IRn →IRm is a function y(x) of the form where and are constants, with i = 1, ,m, and j = 1, n.

Slide 14

 Introducing the vector c ∈ IRm, and the m n matrix A as follows, then, the linear function y(x) can be written as, y = Ax + c. ( Compare it with the expression for a linear function y : IR→ IR, that is, y = ax + c.)

Slide 15

 The product Ax is defined as follows: Exercise: Show that this product satisfies the following properties: A(u + v) = Au + Av, A(cu) = cAu.

Slide 16

 Summary A system of linear equations can be expressed as a linear function, or as a linear combination of the column vectors, respectively, where .

Slide 17

 Theorem 1 Fix and m n matrix , and a vector b ∈ IRm. Then, b ∈ Span  there exist , such that    the echelon form of [A l b] has NO row of the form [0 ...0 l b ≠ 0].
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