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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). 
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

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. 
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 IR^{n} • Definition, Operations, Components. • Linear combinations. • Span. 
Slide 9
Definition, Operations, Components Definition 3 A vector in IR^{n}, 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 ∈ IR^{n} is a linear
combination of p≥ 1 vectors in IR^{n} 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 IR^{n}, 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 : IR^{n} →IR^{m} 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 ∈ IR^{m}, and the m n
matrix A as follows,
then, the linear function y(x) can be written as, 
Slide 15
The product Ax is defined as follows:
Exercise: Show that this product satisfies the
following properties: 
Slide 16
Summary A system of linear equations
can be expressed as a linear function, or as a
linear combination of 
Slide 17
Theorem 1 Fix and m n matrix
, and
a vector b ∈ IR^{m}. 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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