Samenvatting Linear Algebra (X_400649)

I. 1
Systems of linear equations


A linear equation in the variables Xi .
.
. .
.
Xn is an equation that can be written

in the form dik t 92×2 t . . .
tanxn = b.

A
system of linear equations is a collection of one or more linear

linear equations involving the same Variables .
2×1 -
Xzt 1.5×3=0

A solution of the
system is a list (51,52 Sh ) Xi 4×3=-7
-




.
.
.




Of numbers that makes each equation a true statement when the values


51 , . . .

,
Sn are substituted for Xi , . . .
,Xn , respectively .




The set of all possible solutions is called the solution set of the linear

Two linear
system .


systems are called equivalent if
they have the same


solution set .



.




A system of linear equations is said to be consistent if it has either one


solution or
infinitely solutions is inconsistent if it has
many ; a system
no solutions .




The essential information of linear be recorded
a
system can compactly

in called matrix
a
rectangular array augmented
column
column

X, -


2×2 t X3 =
0 I -2 I row I -2 I 0


2×2 0×3 =
0 0 2 -0 0 2 -0 0
-




5×1 -


5×3 =
10 5 0 -5 5 0 -5 10


linear system coefficient matrix matrix
augmented


An
augmented matrix of a system consists of the coefficient matrix

with an added column containing the constants from the right side of the


equations
The size of a matrix tells how many rows and columns it has .




An m x n matrix contains m rows and n columns

The basic strategy to solve a linear system is to replace one system

with an equivalent system that is easier to solve .




Three basic to simplify linear
operations are used a
system :
1. replacement replace one row
by the sum of itself and a multiple

of another row


2. interchange interchange two rows


3. scaling multiply all entries in a row by a nonzero constant

, Two if there is of
matrices are called row
equivalent a sequence elementary
row operations that transform one matrix into the other


Two fundamental questions about a linear system are :




1. IS the system consistent ; that is, does at least one solution exist ?

2 .
If a solution exists ,
is it the only one ; that is ,
is the solution unique ?


1.2 row reduction and echelon forms



A nonzero row or column in a matrix means a row or column that contains

at least one nonzero entry .




A row)
leading entry refers to the leftmost nonzero entry ( in a nonzero

A
rectangular matrix is in echelon form if it has the following properties :




1 all nonzero rows are above rows of all zeros echelon matrix
.



any
2 -3 2 I
2. each leading entry of a row is in a column to the


right of the leading entry of the row above it 0 I -4 0
I
3 all entries in column below 0 0 0
.
a a
leading entry are zeros 2


If a matrix in echelon form satisfies the following additional conditions ,




then it is in reduced echelon form :

I 0 0
2g
4 the leading entry in each nonzero row is I
0 16
0 I
.




is in its column
5. each
leading 1 the only nonzero entry O 0 I 3

It a matrix A is now equivalent to an echelon matrix U ,
reduced echelon matrix


we call U an echelon form of A ;


if u is in reduced echelon form ,
we call U the reduced echelon form of # .




Uniqueness of the reduced echelon form

Each matrix is now equivalent to only one reduced echelon matrix .




A pivot position in a matrix A is a location in A that corresponds to a



leading 1 in the reduced echelon form of A. A pivot column is a column

of A that contains a pivot position .




A pivot is a nonzero number in a pivot position that is used as needed

to create zeros via row operations

pivot columns

*
I 4 5 -




g
-



7
am
* * *



0 4 form : 0
am
2 -6 -6 General * * *


*
pivot 0 0 0 0
am
0 0 -5 0

, row reduction algorithm to produce a matrix in echelon form :




This is
step I
begin with the leftmost nonzero column .
a pivot column .




The pivot position is at the top .




Step 2 select nonzero entry in the pivot column pivot If
a as
necessary
-

.
,




interchange rows to move this entry to the pivot position .




step 3 use row replacement operations to create zeros in all positions

below the pivot .




Step 4 cover cor ignore ) the row
containing the pivot position and


cover all rows ,
if
any ,
above it .

Apply steps I -3 to the


sub matrix that remains .
Repeat the process until there are no



more nonzero rows to
modify .




If we want the reduced echelon form ,
we perform one more step .




Step with the
5 beginning rightmost pivot and working upward and to


the left , create zeros above each pivot . If a pivot is not 1 ,




make it 1 by a scaling operation .




The combination of steps I -4 is called the forward phase of the row


reduction
algorithm . Step 5 is called the backward phase .




In the
following system of equations , the variables X, and Xz are called

basic variables and Xs is called a free variable .
The statement

"
Xz is free "
in the parametric description means that are free
you
to chose value for X3
any



{
.




I 0 -5 I ×,
-


5×3 = 1 Xi =
It 5×3

0 1 I 4 Xz TX3 = 4 Xz= 4 -
Xs

O O O O 0 =
0 Xz is free




whenever a system is inconsistent .
the solution set is empty , even

when the
system has free variables . In this case ,
the solution set


has no parametric representation .




Existence and uniqueness theorem

A linear consistent iff the rightmost column of the
system is augmented
matrix is not a
pivot column .
If a linear
system is consistent , then the

solution set contains either a unique solution or infinitely many
solutions c. when there is at least one free variable )