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The Manga Guide to Linear Algebra by Ltd. Trend-Pro Co., Iroha Inoue, Shin Takahashi

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Bases 145
The vectors of the foowing set do not
form a basis.
1
0
0
1
,
3
1
,
1
2
,
The set
To understand why they don't form a basis,
have a lk at the foowing equation:
x
2
1
2
1 3 x
1
O
= c
1
y
1
y
2
1
0
0
1
+ c
2
3
1
+ c
3
1
2
+ c
4
where is an arbitrary vector in R
2
.
can be formed in many dierent ways
(using dierent choices for c
1
, c
2
, c
3
, and c
4
).
Because of this, the set does not form “a
minimal set of vectors nded to expre
an arbitrary vector in R
m
.”
y
1
y
2
y
1
y
2
146 Chapter 6 More Vectors
Neither of the two vector sets below is able
to describe the vector , and if they can’t
describe that vector, then there's no way that
they could describe “an arbitrary vector in R
3
.”
Because of this, they're not bases.
,
The set
1
0
0
0
1
0
,
The set
1
0
0
,
0
1
0
1
2
0
0
0
1
,
1
0
0
,
0
1
0
0
0
1
Just because a set of vectors is linearly
independent doesn't mean that it forms a basis.
For instance, the set forms a basis,
while the set does not, even though
they're both linearly independent.
,
1
0
0
0
1
0
x
2
x
3
1
1
O
x
1
x
2
x
3
1
1
O
2
x
1
Bases 147
Linear Independence
= c
1
+ c
2
+ … + c
n
a
11
a
21
a
m1
a
12
a
22
a
m2
a
1n
a
2n
a
mn
0
0
0
c
1
= 0
c
2
= 0
c
n
= 0
where the left side is the zero vector of R
m
.
to the equation
if there’s only one solution
We say that a set of vectors is linearly independent
, … ,
a
11
a
21
a
m1
a
12
a
22
a
m2
a
1n
a
2n
a
mn
,A set of vectors forms a basis if there’s only
y
1
y
2
y
m
where the left side is an arbitrary vector in R
m
. And once again, a basis
is a minimal set of vectors needed to express an arbitrary vector in R
m
.
y
1
y
2
y
m
= c
1
+ c
2
+ … + c
n
a
11
a
21
a
m1
a
12
a
22
a
m2
a
1n
a
2n
a
mn
one solution to the equation
, … ,
a
11
a
21
a
m1
a
12
a
22
a
m2
a
1n
a
2n
a
mn
,
Since bases and linear independence are
confusingly similar, I thought I'd talk a bit
about the dierences betwn the two.
Bases
148 Chapter 6 More Vectors
So...
While linear
independence is about
finding a clear-cut path
back to the origin,
We're
linearly
independent!
They're
bases.
They
are.
Ye p.
Exactly!
Not a lot of
people are
able to grasp
the dierence
betwn the
two that fast!
I must say I'm
impreed!
No big deal!
That's a for
tod
Ah, wait a
sec!
bases are about finding
clear-cut paths to any
vector in a given space
R
m
?

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