MTHS 212
LINEÊRE ALGEBRA
LINEAR ALGEBRA
Wat is lineêre algebra?
What is linear algebra?
The branch of mathematics that deals with
the theory of systems of linear equations,
matrices, vector spaces, determinants, and
linear transformations.
The American Heritage® Dictionary of the
English Language, Fourth Edition copyright
©2000 by Houghton Mifflin Company.
Updated in 2009. Published by Houghton
Mifflin Company.
LINEÊRE VERGELYKINGS
LINEAR EQUATIONS
Stelsels lineêre vergelykings
Systems of linear equations
Uitkomste / Outcomes
 Aangevulde matrikse / Augmented matrices
 Ekwivalente stelsels / Equivalent systems
 Elementêre rybewerkings/ Elementary row operations
 Herleiding na rytrapvorm / Transforming to row echelon
form
 Bepaal aantal oplossings van stelsels sowel as oplossings
self / Determine number of solutions as well as solutions
themselves.
 Toepassings / Applications
Definisie / Definition
‘n Lineêre vergelyking in die veranderlikes
A linear equation in the variables x1 ,, xn
is ‘n vergelyking wat geskryf kan word in die form
is an equation that can be written in the form
a1 x1 + a2 x2 +  + an xn =
b
waar b en die koëffisiënte
a1 ,, an
where b and the coefficients
reële of komplekse getalle is wat gewoonlik bekend is.
are real or complex numbers that are usually known in
advance.
Nog definisies / More definitions
‘n Stelsel lineêre vergelykings (of ‘n lineêre stelsel) is
‘n versameling van een of meer lineêre vergelykings
in dieselfde veranderlikes, byvoorbeeld 𝑥𝑥1, … . , 𝑥𝑥𝑛𝑛.
A system of linear equations (or a linear system)
is a collection of one or more linear equations
involving the same variables — say, 𝑥𝑥1, … . , 𝑥𝑥𝑛𝑛.
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Nog definisies / More definitions
‘n Oplossing van die stelsel is ‘n lys (𝑠𝑠1, 𝑠𝑠2, … , 𝑠𝑠𝑛𝑛)
van getalle wat elke vergelyking ‘n waar bewering
maak as die waardes 𝑠𝑠1, … , 𝑠𝑠𝑛𝑛 vervang word in
𝑥𝑥1, … , 𝑥𝑥𝑛𝑛.
A solution of the system is a list (𝑠𝑠1, 𝑠𝑠2, … , 𝑠𝑠𝑛𝑛) of
numbers that makes each equation a true statement
when the values 𝑠𝑠1, … , 𝑠𝑠𝑛𝑛 are substituted for
𝑥𝑥1, … , 𝑥𝑥𝑛𝑛, respectively.
Nog definisies / More definitions
Die versameling van alle moontlike oplossings word
die oplossingsversameling van die lineêre stelsel
genoem.
The set of all possible solutions is called the solution
set of the linear system.
Twee lineêre stelsels word ekwivalent genoem as
hulle dieselfde oplossingsversameling het.
Two linear systems are called equivalent if they have
the same solution set.
Grafiese interpretasie – 2 vgls, 2 onbekendes
Graphical interpretation – 2 eqs, 2 unknowns
Geen oplossing
No solution
Unieke oplossing
Unique solution
Oneindig veel oplossings / Infinitely many solutions
Grafiese interpretasie – 2 of 3 vgls, 2 of 3 onbekendes
Graphical interpretation – 2 or 3 eqs, 2 or 3 unknowns
Verklaring van tipes op vorige skyfie
Explanation of types on previous slide
A.
B.
C.
D.
E.
F.
3 vlakke wat sny in een punt (unieke oplossing)
3 planes intersecting in a single point.
3 vlakke wat sny in 'n lyn (oneindig veel oplossings)
3 planes intersecting in a single point.
3 nie-ewewydige vlakke met geen gemeenskaplike snypunte nie.
3 nonparallel planes with no common points of intersection.
3 vlakke wat saamval (oneindig veel oplossings vir stelsel)
3 planes coinciding (infinitely many solutions for system)
3 vlakke, 2 ewewydig en derde een nie-ewewydig – geen
gemeenskaplike snypunte nie (geen oplossing nie)
3 planes, 2 parallel and third nonparallel – no common points (no sol)
3 ewewydige vlakke nie (geen oplossing nie)
3 parallel planes (no solution)
Verklaring van tipes op vorige skyfie (vervolg)
Explanation of types on previous slide (continued)
G.
H.
I.
J.
K.
L.
Twee ewewydige vlakke (of 3, waarvan 2 saamval – geen oplossing
nie)
2 parallel planes (or 3, of which 2 coincide – no solution)
Twee lyne wat sny in een punt (unieke oplossing)
Two lines intersection in one point (unique solution)
3 lyne, 2 ewewydig en derde nie-ewewydig (geen oplossing nie)
3 lines, 2 parallel and the third nonparallel (no solution)
2 ewewydige lyne (of 3 lyne, waarvan 2 saamval – geen oplossing nie)
2 parallel lines (or 3 lines, of which 2 coincide – no solution)
3 nie-ewewydige lyne wat sny in 'n enkele punt (unieke oplossing)
3 nonparallel lines intersection in a single point (unique solution)
2 of meer lyne wat saamval (oneindig veel oplossings)
2 or more lines coinciding (infinitely many solutions)
Verklaring van tipes op vorige skyfie (vervolg)
Explanation of types on previous slide (continued)
M. 2 vlakke wat sny in 'n lyn (oneindig veel oplossings)
N.
O.
2 planes intersection in a line (infinitely many solutions)
3 ewewydige lyne (geen oplossings)
3 parallel lines (no solutions)
3 nie-ewewydige lyne met geen gemeenskaplike snypunt nie (geen
oplossing nie)
3 nonparallel lines with no common point of intersection (no
solution)
Nog definisies / More definitions


‘n Stelsel lineêre vgls het / A system of lin. eqs has
1. geen oplossing nie, of / no solution, or
2. presies een oplossing /exactly one solution, or
3. oneindigveel opl. / infinitely many solutions.
Ons sê ‘n stelsel lin vgls is oplosbaar of
konsistent as dit een of oneindigveel opl het en
onoplosbaar of inkonsistent as dit geen opl het.
A system of linear equations is said to be
consistent if it has either one solution or infinitely
many solutions and inconsistent if it has no
solution.
Afspraak / Agreement
Om stelsels vergelykings op te los, skryf ons hulle in
aangevulde matriksvorm, byvoorbeeld ons skryf / To
solve systems of equations, we write them in augmented
matrix form, for example we write
as
Strategie / Strategy
Ons vervang ‘n stelsel met ‘n ekwivalente
eenvoudiger stelsel deur elementêre rybewerkings
uit te voer.
We replace a system with an equivalent simpler
system by means of elementary row operations.
ELEMENTÊRE RYBEWERKINGS
ELEMENTARY ROW OPERATIONS
Ons kry die volgende elementêre rybewerkings:
Elementary row operations include the following:
1. (Vervanging) Vervang een ry met die som van
homself en ‘n veelvoud van ‘n ander ry.
(Replacement) Replace one row by the sum of
itself and a multiple of another row.
2. (Ruiling) Ruil twee rye
(Interchange) Interchange two rows.
3. (Skalering) Vermenigvuldig al die
inskrywings in ‘n ry met ‘n nie-nul konstante.
(Scaling) Multiply all entries in a row by a
nonzero constant.
Slide 1.1- 18
© 2012 Pearson Education, Inc.
Definisie / Definition
Twee matrikse word ry-ekwivalent genoem as daar
‘n ry van elementêre rybewerkings is wat een
matriks verander in die ander.
Two matrices are called row equivalent if there is a
sequence of elementary row operations that
transforms one matrix into the other.