Basis of a Set in Linear Algebra
Confirming that
a set is a Basis
Intuitive Way
•
Definition: A
basis
B for V is an
independent
generation set
of V.
Is
C
a basis of
V
?
Independent?
yes
Generation set?
difficult
generates V
Another way
•
Given a subspace V, assume that we already know that dim
V = k.
Suppose S is a subset of V with k vectors
If S is independent
S is basis
If S is a generation set
S is basis
Is
C
a basis of
V
?
Independent?
yes
C
is a basis of
V
Dim V = 2
(parametric representation)
C
is a subset of V with 2 vectors
Find a basis for V
Another way
Assume that dim V = k.
Suppose
S is a subset of V with k vectors
By the extension theorem, we can add more
vector into S to form a basis.
However, S already have k vectors, so it is already
a basis.
By the reduction theorem, we can remove some
vector from S to form a basis.
However, S already have k vectors, so it is already
a basis.
Example
•
Is
B
a basis of
V
?
Dim V = ?
3
Independent set in V?
yes
B
is a basis of
V
.
Example
•
Is
B
a basis of
V
= Span
S
?
B
is a subset of V with 3
vectors
B
is a basis of
V
.
A basis for a vector space V is an independent generating set. There are intuitive ways to confirm if a set is a basis, such as checking if it is independent and generates V. The dimension of V helps determine if a subset is a basis. Examples and methods like the extension theorem are explored to find or verify bases of sets in linear algebra.
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Presentation Transcript
Confirming that a set is a Basis
Intuitive Way Definition: A basis B for V is an independent generation set of V. Is Ca basis of V? Independent? yes Generation set? difficult generates V
Another way Find a basis for V Given a subspace V, assume that we already know that dim V = k. Suppose S is a subset of V with k vectors If S is independent If S is a generation set S is basis S is basis Is Ca basis of V? Dim V = 2 (parametric representation) C is a subset of V with 2 vectors Cis a basis of V Independent? yes
Assume that dim V = k. Suppose S is a subset of V with k vectors Another way If S is independent S is basis By the extension theorem, we can add more vector into S to form a basis. However, S already have k vectors, so it is already a basis. If S is a generation set S is basis By the reduction theorem, we can remove some vector from S to form a basis. However, S already have k vectors, so it is already a basis.
Example Is Ba basis of V? Independent set in V? yes Bis a basis of V. Dim V = ? 3
Example B is a subset of V with 3 vectors Is Ba basis of V = Span S? 0 0 2/3 1/3 2/3 0 1 0 0 0 0 1 0 0 ??= ???? = 3 1 0 1 0 0 0 0 1 0 0 0 0 1 0 ??????????? ??= Bis a basis of V.
