ENGG2013 Unit 13
Basis
Feb, 2011.
Question 1
• Find the value of c1 and c2 such that
kshum
ENGG2013
2
Question 2
• Find the value of c1 and c2 such that
kshum
ENGG2013
3
Question 3
• Find c1, c2, c3 and c4 such that
kshum
ENGG2013
4
Basis: Definition
• For any given vector
in
if there is one and only one choice for the
coefficients c1, c2, …,ck, such that
we say that these k vectors form a basis of
kshum
ENGG2013
.
5
Example
•
form a basis of
.
• Another notation is:
is a basis of
.
1
1
kshum
ENGG2013
6
Example
•
form a basis of
.
• Another notation is:
is a basis of
.
2
2
kshum
ENGG2013
7
Non-Example
•
is not a basis of
.
1
1
kshum
ENGG2013
8
Alternate definition of basis
• A set of k vectors
is a basis of
if the k vectors satisfy:
1. They are linear independent
2. The span of them is equal to
(this is a short-hand of the statement that:
every vector in
can be written as a linear
combination of these k vectors.)
kshum
ENGG2013
9
More examples
•
is a basis of
3
3
kshum
ENGG2013
10
Question
• Is
a basis of
z
1
1
x
y
kshum
ENGG2013
11
Question
• Is
a basis of
?
z
1
1
1
x
y
kshum
ENGG2013
12
Question
• Is
a basis of
?
z
1
1
2
3
x
y
kshum
ENGG2013
13
Question
• Is
a basis of
?
z
2
1
1
x
y
kshum
ENGG2013
14
Question
• Is
a basis of
?
z
2
1
1
x
y
kshum
ENGG2013
15
Fact
• Any two vectors in
do not form a basis.
– Because they cannot span the whole
• Any four or more vectors in
basis
.
do not form a
– Because they are not linearly independent.
• We need exactly three vectors to form a basis
of
.
kshum
ENGG2013
16
A test based on determinant
• Somebody gives you three vectors in
.
• Can you tell quickly whether they form a
basis?
kshum
ENGG2013
17
Theorem
This theorem generalizes
to higher dimension naturally.
Just replace 3x3 det by nxn det
Three vectors in
form a basis if and only if the determinant
obtained by writing the three vectors together is non-zero.
Proof:  Let the three vectors be
Assume that they form a basis.
In particular, they are linearly independent. By definition, this means
that if
then c1, c2, and c3 must be all zero.
By the theorem in unit 12 (p.17) , the determinant
kshum
ENGG2013
is nonzero.
18
The direction  of the proof
• In the reverse direction, suppose that
• We want to show that
1. The three columns are linearly independent
2. Every vector in
can be written as a linear
combination of these three columns.
kshum
ENGG2013
19
The direction  of the proof
1. Linear independence: Immediate from the theorem
in unit 12 (8  3).
2. Let
be any vector in
.
We want to find coefficients c1, c2 and c3 such that
Using (8  1), we know that we can find a left
inverse of
. We can multiply by the left
inverse from the left and calculate c1, c2, c3.
kshum
ENGG2013
20
Example
• Determine whether
form a basis.
• Check the determinant of
kshum
ENGG2013
21
Summary
• A basis of contains the smallest number of
vectors such that every vector can be written
as a linear combination of the vectors in the
basis.
• Alternately, we can simply say that: A basis of
is a set of vectors, with fewest number of
vectors, such that the span of them is
.
kshum
ENGG2013
22