Summary

Summary Mathematics 214 Linear Algebra Summaries

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Course
Mathematics 214

Institution
Stellenbosch University (SUN)

Linear Algebra summaries for Mathematics 214 covering all the work done in class for the whole syllabus, weeks 1-12. These summaries include all the work needed for the A1, A2 / A3. These summaries cover all topics in the work plan as well as all examples and theorems discussed in class, with ex...

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Uploaded on June 24, 2023
Number of pages 72
Written in 2022/2023
Type Summary

mathematics 214 linear algebra

Institution
Stellenbosch University (SUN)
Course
Mathematics 214

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Revision of linear algebra summaries are subject to
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Sharing is not permitted.
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-

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