Recap
Consider a linear system of m equations in n variables with
augmented matrix Alb
The system is inconsistent the last non-zero row of a
row echelon form of AID has the form 0 0 O c for
some non-zero c E IR.
If the system is consistent, it has a unique solution
every variable in the system is leading, i.e
every column
of a row echelon form of A contains a leading entry.
Otherwise, the system has infinitely many solutions.
Equivalently
Inconsistent
rank(A) rank Alb
Unique so
rank(A) rank(LAlb) n
Infinitely many sols
rank A
rank [Alb) n.
Example 2.10: Consider the linear system
where a and b are unknown real numbers. Determine all con
ditions on a and b so that the system has
(1) No solutions.
(2) A unique solution.
(3) Infinitely many solutions.
Solution: (Warning: Don't divide by unknown numbers, or
multiply a row by unknown numbers like a or b!!!)
3 2
2
rz-r
1 2 4 3
0 1
3 a b
O 2 a-3 b-2
D 0 a-5 b-4
13-2m2
(1) No solution a -S o and b-4 o I ast row
0 0 0
a 5 and b 4
(2) Unique solution consistent but no free variable
b can tnke an
VA twe
Recap Consider a linear system of m equations in n variables with augmented matrix Alb The system is inconsistent the last non-zero row of a row echelon form of AID has the form 0 0 O c for some non-zero c E IR. If the system is consistent, it has a unique solution every variable in the system is leading, i.e every column of a row echelon form of A contains a leading entry. Otherwise, the system has infinitely many solutions. Equivalently Inconsistent rank(A) rank Alb Unique so rank(A) rank(LAlb) n Infinitely many sols rank A rank [Alb) n. Example 2.10: Consider the linear system where a and b are unknown real numbers. Determine all con ditions on a and b so that the system has (1) No solutions. (2) A unique solution. (3) Infinitely many solutions. Solution: (Warning: Don't divide by unknown numbers, or multiply a row by unknown numbers like a or b!!!) 3 2 2 rz-r 1 2 4 3 0 1 3 a b O 2 a-3 b-2 D 0 a-5 b-4 13-2m2 (1) No solution a -S o and b-4 o I ast row 0 0 0 a 5 and b 4 (2) Unique solution consistent but no free variable b can tnke an VA twe(3) Infinitely many solutions consistent with at least one
free variable
a and
Theorem 2.11: A consistent linear system which involves
re variables than equations has infinitely many solutions.
Proof: leading variables
non -zero rows of re
total rows
re
tion equations
assump
totul 4 variables
Must be at least one
free variable
To apply this theorem, we need examples of systems which are
always consistent.
Definition: A system of linear equations whose constant
terms are all equal to 0 is said to be homogeneous
Example: The following linear system is homogeneous
4r1 3r2
Observation: A homogeneous linear system is always con
sistent, because the vector is clearly a solution of any such
system (we call this the trivial solution
By the above dis
cussion, if a homogeneous linear system has any
non-trivial
lutions, then it has infinitely many solutions.
By Theorem 2.11, we have the following criterion for the exis-
tence of infinitely many solutions:
Theorem 2.12: (Solutions of homogeneous systems)
A homogeneous linear system which involves more variables
than equations has infinitely many solutions.
Back to geometry
The methods developed above allow us to address some basic
geometric problems which arose in Chapter I
Example 2.13: The planes P1 2y z 3 and
P2 2 3y 52 1 in R3 intersect in a line l. Find a
vector equation of that line l
Solution: The intersechon points
an
Pr correspond to He
Solutions
row
2 is a tre variable
(3) Infinitely many solutions consistent with at least one free variable a and Theorem 2.11: A consistent linear system which involves re variables than equations has infinitely many solutions. Proof: leading variables non -zero rows of re total rows re tion equations assump totul 4 variables Must be at least one free variable To apply this theorem, we need examples of systems which are always consistent. Definition: A system of linear equations whose constant terms are all equal to 0 is said to be homogeneous Example: The following linear system is homogeneous 4r1 3r2 Observation: A homogeneous linear system is always con sistent, because the ve
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