elementary linear Need solution for 11 Verify that rankA ra

Verify that rank(A) = rank(A^T) A = [1 2 4 0 -3 1 5 2 -2 3 9 2] Find an equation relating nullity(M) and nullityM7\") for the matrix in Exercise 10. Find an equation relating nullity(A) and nullity (A^T) for a general m times x n matrix.

Solution

An mxn matrix A defines a linear transformation from Rⁿ to R^m.

A^T defines a linear transformation from R^m to Rⁿ.

We have the relations

                                           n = Nullity (A)+ Rank (A)............................(1)

                                           m= Nullity (A^T) + Rank (A^T).........................(2)

(a) In this case m=3, n=4. (1)-(2) gives

                                             Nullity (A)-Nullity(A^T) = 1...........................(3)

(b) Rank (A) = Rank (A^T) is always true, so in general

                                            Nullity (A)-Nullity(A^T) =n-m........................(4)