Summary Mathematics 214 Linear Algebra Summaries

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linear equations
systems of


· matrices and augmented matrices

· Gaussian reduction:row multiplication;add subtractrows,swap rows

·

leading entries:first
non-zero
entry
·

consistent/inconsistentsystems
pow-echelon form:
·
free variables all a
of matrix's
non-zero rows
·
the general solution
have a pivot:a
Linear combinations non-zero
entry
such thatthe
entries
to its leftand below
are all zero.

Systems recap



S
2
2x1 -

3712 573
+

2
=


7
2 -
3 S To solve a
system:
3 13 ① augmented
1
x x2
+ -
x3
=
I -

Set
up matrix

② now reduce to row echelon form

⑤ Detect
consistency
52 -..- 13' ④
2 3 R, R2 Detectfree variables
-




35 2
13 -2 ⑤
-

I 1
solve
-


-



system

R2-2 RI -, -
1 3

-
⑧ -

57-4-

↓

x1 32
+ -

713 3 =




-
972 7x3
+
= -
4


Set73:t

Then -
Siz: -
4-7t

parametric equation
xz
4 It
+
=




5+)
and
x 3
= -

x2 x3 +
3
=
-


(! +

t
+

, Types solutions
of




①
Independent
· consistent(atleast
one solution)
·


unique solution

· row-reduced matrixhas the same no. of non-zero rows as variables

·
LHS is
usually the
identity matrix

there mustbe least variables
at as
many equations as
·




M




Independentsystems produce different
lines. They have one solution. ·
unique
I solution

independent:exactly
consistent one solution Ac 8
= Where 8
= differentlines




② Dependent
·
consistent

many solutions
·




in form it because free variables)
write answer
parametric as
parameter, of
·




·
row-reduced has
matrix more variables than non-zero rows

there doesn'thave to be a row of zeros, butthere usually is
·




·

possibly less equations than variables (no. variables, no equations)
·



equations have the same
slope and same y-intercepts (same line)


③ Inconsistent

·
no solution (null-set/empty set: 1)
· row-reduced has
matrix a row zeros
of on the LHS, but
a non-zero constant

on the RHS




Ac
* b
= is only if
consistent

b lies in the span A.
of


(more on this later)

, Activity
- -




Augmented
I
1. 0 I I matrix

12-12
Inconsistent (no solutions)
- 12
I so, I
=>
0 3
12
-




e.G. or



-0 1, I.



-> x+2x2
-


x3 2
=




x3 1 PIVOT: firstnon-zero
xz
entry
+
=




each
of row when matrix

setx3
t, then x2 1 t is in row echelon form
-
=
=




x1 2
=
-
2(1 -

t) t
+




=


2 -
2 +
2t t
+




3t
-
FREE VARIABLE:

variable thatdoes not

2. -I -
I .-, correspond to a pivotcolumn
I I I I 3 scolumn where pivotappears)
x,(2 73
2
O I 0 I
-
2 -
1 ↳


0-82-3
- -




I . . 1l x, and2 correspond to
. .




2 but
⑧ 20 2
pivotcolumns, (3

0000 2 Inconsistent is a free variable
-
-




-
I, 32 73 -
- -




⑪


I
3. x1 (z
+
x3
+
1
=
I I 11 I I I

000 O
2x1 2x2 +
2x3 2
+
=

-> 2 22 2 =>

3x1 +
3x2 3x3 3 +
=

3 33 3 o 000

4x2 4x3 4 L 4 000 ⑧
4x1 + +
=
4 4 -




are
-
-




x2 and as free variables

,
-




L. i I I I


2 2 2 3
2
-
-




I I
I

i
- I I I


⑧ 0 oo cinconsistent)

, Homogenous vs non-homogeneous systems



Homogenous systems
thatalways have atleastI solution
·

systems
·
Alinear system called
is homogenous the
if term
constant each
in
equation is zero.


Has either:


·
a
unique solution

or
infinitely many solutions
·




need
, no. variables
of to exceed the no.
equations
of




->




-...
I ①
1.
homogenous has leastI
at solution)
21
-

10
- -




-




-.-2,
R2: R2 -2R,
05-3-10
free variables:
correspond to columns thatdon'thave leadingentries

LetC(3 = t


Let<4 S
=




572-3t -

3 0 =




S2
3t 55
= +




-t Es
x1
-




xz
=
8 +
+


Is
:x
2(zt 15) t 5 0
+ + +
- =




33 t


-
34, -
S
-




:(
It S
=
-




,
-




3. a.) ri vz - -
+




7

10


-
S-