Dimension of Basis in Linear Algebra
Dimension of Basis
Hung-yi Lee
李宏毅
Col A
=
Range
•
Basis: The pivot columns of A form a basis for Col A.
•
Dimension:
Col
A
= Span
pivot columns
pivot columns
Dim (Col A)
= number of pivot columns
= rank A
Rank A (revisit)
Row A
•
Basis: Nonzero rows of RREF(A)
•
Dimension:
a basis of
Row
R
Row A = Row R
(The elementary row operations
do not change the row space.)
= a basis of
Row
A
Dim (Row A)
= Number of Nonzero rows
= Rank A
Rank A (revisit)
= Dim (Row A)
= Dim (Col A
T
)
Rank A = Rank A
T
•
Proof
= Dim (Row A)
= Dim (Col A
T
)
Rank A
Rank A
= Dim (Col A)
= Rank A
T
Null A
•
Basis:
•
Solving Ax = 0
•
Each free variable in the parametric representation of
the general solution is multiplied by a vector.
•
The vectors form the basis.
Example 2, P256
Basis
(free)
(free)
Null A
•
Basis:
•
Solving Ax = 0
•
Each free variable in the parametric representation of
the general solution is multiplied by a vector.
•
The vectors form the basis.
•
Dimension:
Dim (Null A)
= number of free variables
= Nullity A
= n - Rank A
Dimension Theorem
Dim of Range
Dim of Null
Dim of Domain
A
A
range
+
=
Dim (Col A)
Dim (Null A)
If A is mxn
=Rank A
=n - Rank A
Dim (R
n
) =n
This informative content delves into the dimension of basis in linear algebra, covering essential concepts such as range basis, pivot columns, rank, row basis, and more. It explains how to determine the dimension of column space and the range of a matrix, along with exploring examples and the null space basis. The dimension theorem is also discussed, providing insights into the dimensions of column space, null space, range, and domain. Rich with visual aids, this content serves as a valuable resource for understanding fundamental linear algebra principles.
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Presentation Transcript
Dimension of Basis Hung-yi Lee
Col A = Range Basis: The pivot columns of A form a basis for Col A. Col A = Span pivot columns pivot columns Dimension: Dim (Col A) = number of pivot columns = rank A
Rank A (revisit) Maximum number of Independent Columns Number of Pivot Columns Number of Non-zero rows Number of Basic Variables Dim (Col A): dimension of column space Dimension of the range of A
Row A Basis: Nonzero rows of RREF(A) RREF R= Row A = Row R (The elementary row operations do not change the row space.) a basis of Row R = a basis of Row A Dimension: Dim (Row A) = Number of Nonzero rows = Rank A
Rank A (revisit) Maximum number of Independent Columns Number of Pivot Column Number of Non-zero rows Number of Basic Variables Dim (Col A): dimension of column space = Dim (Row A) = Dim (Col AT) Dimension of the range of A
Rank A = Rank AT Proof Rank A = Dim (Col A) Rank A = Dim (Row A) = Dim (Col AT) = Rank AT
Example 2, P256 2 1 5 3 1 0 0 0 0 1 0 0 1 0 0 1 0 1 4 3 1 1 0 1 0 5 1 Null A 5 0 0 ? = ? = 2 0 5 2 2 1 5 2 3 10 Basis: Solving Ax = 0 Each free variable in the parametric representation of the general solution is multiplied by a vector. The vectors form the basis. ?1 ?2 ?3 ?4 ?5 ?1= ?3 ?5 ?2= 5?3 4?5 ?3= ?3 1 5 1 0 0 1 4 0 2 1 ?1+ ?3+ ?5= 0 ?2 5?3+ 4?5= 0 ?4 2?5= 0 = ?3 + ?5 (free) ?4= 2?5 ?5= ?5 (free) Basis
Null A Basis: Solving Ax = 0 Each free variable in the parametric representation of the general solution is multiplied by a vector. The vectors form the basis. Dimension: Dim (Null A) = number of free variables = Nullity A = n - Rank A
Dimension Theorem Dim (Col A) If A is mxn Dim (Null A) =n - Rank A Dim (Rn) =n =Rank A Dim of Null Dim of Range + = Dim of Domain A A range
