Understanding Rank and Nullity in Linear Algebra
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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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
