Linear Dependent and Independent Vectors
How many solutions?
Hung-yi Lee
Reference
•
Textbook: Chapter 1.7
Review
No
solution
YES
NO
Given a system of linear equations with
m equations and n variables
Have
solution
H
ow many solutions?
Today
No
solution
YES
Rank A = n
Nullity A = 0
Unique solution
Rank A < n
Nullity A > 0
Infinite solution
NO
Given a system of linear equations with
m equations and n variables
Other cases?
Dependent and
Independent
(
依賴的、不獨立的
)
(
獨立的、自主的
)
Definition
Find one
Obtain many
unique
Dependent and Independent
Dependent or Independent?
Dependent or Independent?
Dependent and Independent
Dependent or
Independent?
Any set contains zero vector would be linear dependent
Zero vector is the linear combination of any other vectors
How about a set with only one vector?
Dependent and Independent
Intuition
•
Intuitive link between dependence and the number
of solutions
Infinite
Solution
Dependent:
Once we have solution, we have infinite.
Proof
•
Columns of
A
are
dependent
→ If
A
x
=
b
have
solution, it will have Infinite Solutions
•
If
A
x
=
b
have Infinite solutions → Columns of
A
are
dependent
Proof
Homogeneous linear equations
infinite
Based on the definition
Proof
•
Columns of
A
are
dependent
→ If
A
x
=
b
have
solution, it will have Infinite solutions
•
If
A
x
=
b
have Infinite solutions → Columns of
A
are
dependent
Non-zero
Rank and Nullity
Rank and Nullity
•
The
rank
of a matrix is defined as the maximum
number of
linearly independent columns
in the
matrix.
•
Nullity
= Number of columns -
rank
Rank and Nullity
•
The
rank
of a matrix is defined as the maximum
number of
linearly independent columns
in the
matrix.
•
Nullity
= Number of columns -
rank
Rank and Nullity
•
The
rank
of a matrix is defined as the maximum
number of
linearly independent columns
in the
matrix.
•
Nullity
= Number of columns -
rank
If A is a mxn matrix:
Rank A = n
Nullity A = 0
Columns of A are
independent
Conclusion
No
solution
YES
Rank A = n
Nullity A = 0
Unique solution
Rank A < n
Nullity A > 0
Infinite solution
NO
Conclusion
Rank A = n
Nullity A = 0
YES
NO
YES
NO
No
solution
Unique
solution
YES
NO
No
solution
Infinite
solution
Question
•
True or False
•
If the columns of A are linear independent, then Ax=b
has unique solution.
•
If the columns of A are linear independent, then Ax=b
has at most one solution.
•
If the columns of A are linear dependent, then Ax=b has
infinite solution.
•
If the columns of A are linear independent, then Ax=0
(homogeneous equation) has unique solution.
•
If the columns of A are linear dependent, then Ax=0
(homogeneous equation) has infinite solution.
New Question:
Why Homogeneous name homogeneous? Does it have applications?
Extension:
intuitive definition to another definition => should I proof it?
Why only one or inifintie soluiton?
Intuitive link of infiitnie solution and independent
Question:
Example for 冗員
the correct definition of rank
dependent is defined on a set or a vector? On set
In linear algebra, when exploring systems of linear equations and vector sets, it is crucial to distinguish between linear dependent and independent vectors. Linear dependence occurs when one vector can be expressed as a combination of others, leading to various solutions or lack thereof in the given system. This concept plays a fundamental role in determining the solutions of a system of linear equations.
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Presentation Transcript
How many solutions? Hung-yi Lee
Reference Textbook: Chapter 1.7
Given a system of linear equations with m equations and n variables Review ? ??? ?? ?:? ? Is ? in the span of the columns of ?? Is ? a linear combination of columns of ?? YES NO No Have solution solution How many solutions?
Given a system of linear equations with m equations and n variables Today ? ?? ? ?? ?:? ? Is ? in the span of the columns of ?? Is ? a linear combination of columns of ?? YES NO The columns of ? are independent. The columns of ? are dependent. No solution Rank A = n Rank A < n Nullity A = 0 Unique solution Nullity A > 0 Infinite solution Other cases?
() Dependent and Independent ( )
Definition A set of n vectors ?1,?2, ,?? is linear dependent If there exist scalars ?1,?2, ,??, not all zero, such that ?1?1+ ?2?2+ + ????= ? Obtain many Find one A set of n vectors ?1,?2, ,?? is linear independent ?1?1+ ?2?2+ + ????= ? Only if ?1= ?2= = ??= 0 unique
Dependent and Independent Linear Dependent Given a vector set, {a1, a2, , an}, if there exists any aithat is a linear combination of other vectors 4 12 6 10 30 15 Dependent or Independent? , 7 6 3 3 1 8 3 , , Dependent or Independent? 11 6
Dependent and Independent Linear Dependent Given a vector set, {a1, a2, , an}, if there exists any aithat is a linear combination of other vectors Dependent or Independent? Zero vector is the linear combination of any other vectors Any set contains zero vector would be linear dependent How about a set with only one vector?
Dependent and Independent Linear Dependent Given a vector set, {a1, a2, , an}, if there exists any aithat is a linear combination of other vectors 2??+ ??+ 3??= ? 2??+ ??= 3?? 2 3 ?? = 3?? + 4?? ?? 3?? 4?? = ? ??+ 1 ??= ?? 3 Given a vector set, {a1, a2, , an}, there exists scalars x1, x2, , xn, that are not all zero, such that x1a1+ x2a2+ + xnan= 0.
Dependent: Once we have solution, we have infinite. Intuition Intuitive link between dependence and the number of solutions 6 1 7 3 8 11 3 3 6 ?3 12 6 3 3 3 6 6 3 3 3 12 ?1 ?2 7 6 3 3 1 8 3 14 22 1 + 1 = 11 6 = ?1 ?2 ?3 7 1 1 1 1 8 14 22 12 = 1 + 1 + 1 = 11 ?1 ?2 ?3 1 8 14 22 2 2 0 Infinite Solution 2 + 2 = =
Proof Columns of A are dependent If Ax=b have solution, it will have Infinite Solutions If Ax=b have Infinite solutions Columns of A are dependent
Proof Homogeneous linear equations ?1 ?2 ?? ?? ? = ?1 ?2 ?? = ? ? = (always having ? as solution) Based on the definition A set of n vectors ??,??, ,?? is linear dependent ?? = ?have non- zero solution infinite A set of n vectors ??,??, ,?? is linear independent ?? = ?only have zero solution
Proof Columns of A are dependent If Ax=b have solution, it will have Infinite solutions We can find non-zero solution u such that ?? = ? ? ? + ? = b ? + ? is another solution different to v There exists v such that ?? = b If Ax=b have Infinite solutions Columns of A are dependent ?? = b ?? = b ? ? ? = ? Non-zero ? ?
Rank and Nullity The rank of a matrix is defined as the maximum number of linearly independent columns in the matrix. Nullity = Number of columns - rank 3 7 0 2 9 0 1 0 2 1 2 3 3 6 9 10 20 30 0 0 0 0 0 0 0 0 0
Rank and Nullity The rank of a matrix is defined as the maximum number of linearly independent columns in the matrix. Nullity = Number of columns - rank 1 2 3 6 4 8 0 0 3 5 5 2 6
Rank and Nullity The rank of a matrix is defined as the maximum number of linearly independent columns in the matrix. Nullity = Number of columns - rank If A is a mxn matrix: Rank A = n Columns of A are independent Nullity A = 0
Conclusion ? ?? ? ?? ?:? ? Is ? in the span of the columns of ?? Is ? a linear combination of columns of ?? YES NO The columns of ? are independent. The columns of ? are dependent. No solution Rank A = n Rank A < n Nullity A = 0 Unique solution Nullity A > 0 Infinite solution
Conclusion The columns of ? are independent. ?:? ? ? ??? ?? Rank A = n Nullity A = 0 NO YES Is ? a linear combination of columns of ?? Is ? a linear combination of columns of ?? Is ? in the span of the columns of ?? Is ? in the span of the columns of ?? NO No YES NO No YES Unique solution Infinite solution solution solution
Question True or False If the columns of A are linear independent, then Ax=b has unique solution. If the columns of A are linear independent, then Ax=b has at most one solution. If the columns of A are linear dependent, then Ax=b has infinite solution. If the columns of A are linear independent, then Ax=0 (homogeneous equation) has unique solution. If the columns of A are linear dependent, then Ax=0 (homogeneous equation) has infinite solution.
