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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

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.


• 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

A system of linear equations can be written as a vector equation:

Slide 12


Definition 5 Given p vectors in IRn, denote by
Span the set of all linear combinations of .


• 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


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


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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