Engineering Analysis Chapter 7 1 RANK AND NULLITY

Engineering Analysis Chapter 7 1

RANK AND NULLITY 2

RANK AND NULLITY 3

RANK AND NULLITY EXAMPLE Reducing A to row-echelon form, we obtain 4

RANK AND NULLITY the nonzero row vectors of R form a basis for the row space of R and hence form a basis for the row space of A. These basis vectors are The first, third, and fifth columns of R contain the leading 1's of the row vectors, so 5

RANK AND NULLITY form a basis for the column space of R; thus the corresponding column vectors of A—namely form a basis for the column space of A. each have three basis vectors; that is, both are three-dimensional. It is not accidental that these dimensions are the same; it is a consequence of the following general result 6

RANK AND NULLITY THEOREM DEFINITION The common dimension of the row space and column space of a matrix A is called the rank of A and is denoted by rank(A); the dimension of the nullspace of A is called the nullity of A and is denoted by nullity(A). 7

RANK AND NULLITY EXAMPLE Rank and Nullity of 4 x 6 a Matrix Find the rank and nullity of the matrix Solution The reduced row-echelon form of A is (1) 8

RANK AND NULLITY • Since there are two nonzero rows (or, equivalently, two leading 1's), the row space and column space are both two-dimensional, so rank (A) = 2. • To find the nullity of A, we must find the dimension of the solution space of the linear system Ax =0. • This system can be solved by reducing the augmented matrix to reduced row-echelon form. The resulting matrix will be identical to 1, except that it will have an additional last column of zeros, and the corresponding system of equations will be or, on solving for the leading variables 9

RANK AND NULLITY It follows that the general solution of the system is or, equivalently, Because the four vectors on the right side of 3 form a basis for the solution space, nullity (A) =4. 10

RANK AND NULLITY THEOREM 11

RANK AND NULLITY THEOREM 12

RANK AND NULLITY EXAMPLE The Sum of Rank and Nullity The matrix has 6 columns, so This is consistent with previous Example, where we showed that 13

RANK AND NULLITY EXAMPLE Number of Parameters in a General Solution Find the number of parameters in the general solution of Ax=0 if A is a 5 x 7 matrix of rank 3. Solution Thus there are four parameters. 14

RANK AND NULLITY 15

RANK AND NULLITY EXAMPLE Maximum Value of Rank for a Matrix If A is a 7 x 4 matrix, then the rank of A is at most 4, and consequently, the seven row vectors must be linearly dependent. If A is a 4 x 7 matrix, then again the rank of A is at most 4, and consequently, the seven column vectors must be linearly dependent. 16

RANK AND NULLITY EXAMPLE Find a basis for the row space of the matrix A Also, compute the row rank of A. 17

RANK AND NULLITY Since the leading 1 's in (2) occur in columns 1, 2 , and 3, we conclude that the first three rows of A form a basis for the row space of A. That is, is a basis for the row space of A. The row rank of A is 3. 18

RANK AND NULLITY EXAMPLE Find a basis for the column space of the matrix A and compute the column rank of A Solution whose augmented matrix is [ A ! 0]. Transforming this matrix to reduced row echelon form, we obtain Since the leading 1 's occur in columns 1, 2, and 4, we conclude that the first, second, and fourth columns of A form a basis for the column space of A. That is , 19

RANK AND NULLITY is a basis for the column space of A. The column rank of A is 3. 20

RANK AND NULLITY EXAMPLE find the rank and nullity of matrix A Solution When A is transformed to reduced row echelon form, we get Then rank A = 3 and nullity A = 2. 21

RANK AND NULLITY EXAMPLE let Transforming A to reduced row echelon form, we find that so we conclude the following: • Rank A= 2, and nullity A = 1 From the reduced row echelon form matrix that A has been transformed to, we see that every solution to the homogeneous system Ax = 0 is of the form 22

RANK AND NULLITY where r is an arbitrary constant, so the solution space of this homogeneous system, or the null space of A, is a line passing through the origin. Moreover, the dimension of the null space of A, or the nullity of A, is 1. 23