Rank and Nullity in Linear Algebra
Rank and Nullity
秩
零度
Rank and Nullity
•
The
rank
of a matrix is defined as the maximum
number of
linearly independent columns
•
Nullity
= Number of columns -
rank
maximum
Independent set
所有
登庸武將
最大沒人
耍廢軍團
Rank and Nullity
以下是最土炮的作法
dependent
dependent
dependent
dependent
independent
independent
independent
Rank and Nullity
Assume the three
columns are independent
If A is a mxn matrix (n columns)
Rank A = n
Nullity A = 0
Columns of A are
independent
Summary
No
solution
YES
Rank
A = n
Nullity
A = 0
Unique solution
Rank
A < n
Nullity
A > 0
Infinite solution
NO
=
=
次序、規則、條理。如：「秩序」。
The rank of a matrix is the maximum number of linearly independent columns, while the nullity is obtained by subtracting the rank from the number of columns. Linearly independent columns form the basis for the rank of a matrix, helping determine if a given matrix has a unique solution, infinite solutions, or no solution. Explore the concepts further through examples and discussions on column independence and linear combinations.
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Presentation Transcript
Rank and Nullity The rank of a matrix is defined as the maximum number of linearly independent columns Nullity = Number of columns - rank ???? ? = 4 ??????? ? = 2 ? = maximum Independent set
Rank and Nullity 1 2 3 3 6 9 10 20 30 ???? ? =?,??????? ? =? ? = 1 2 3 3 6 9 10 20 30 1 2 3 3 6 9 1 2 3 10 20 30 3 6 9 10 20 30 , , , , , dependent dependent dependent dependent ???? ? = 1 1 2 3 3 6 9 10 20 30 ??????? ? = 2 independent independent independent
Rank and Nullity 3 7 0 2 9 0 1 0 2 ???? ? =?3 A = ??????? ? =?0 Assume the three columns are independent If A is a mxn matrix (n columns) Rank A = n Columns of A are independent Nullity A = 0
???? ? =?1 ? =1 3 6 4 8 2 ??????? ? =?2 0 0 0 0 0 0 0 0 0 ???? ? =?0 0 0 0 ? = ??????? ? =?3 ? =0 3 5 ? =5 0 2 ???? ? =?1 ???? ? =?1 ??????? ? =?0 ??????? ? =?1
Summary ? ?? ? ?? ?:? ? 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
