The linear transformation TRn Rm is given by Tx Ax find the

The linear transformation T:Rⁿ - R^m is given by T(x) =Ax, find the nullity and rank of T, determine whether T is one to one, onto, or neither. Explain your answers.

a) A= ( 3x3 matrix ) 1   2    0

                                 0    1    1

                                 0    0   1

b) A= (3x2)       1     2

                          0     1

                          0     0

c)   A= (2x3)      1      2      0

                          0      1    -1        

d) A=   (3x3)       1    2    0

                             0    1    1

                             0     0    0

Solution

a) A= ( 3x3 matrix ) 1   2    0

                                 0    1    1

                                 0    0   1

Rank of A = 3

Determinant value = 1 hence has unique solution

and invertible

Hence one to one and onto

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b) A= (3x2)       1     2

                          0     1

                          0     0

Rank =2

and kernel would be (0,0,z)

Nullity is all vectors in xy plane

c) Rank =2

x+2y=0 y+z =0

x-2z =0

Not one to one or onto

Nullity is all vectors of the form

(2,-1,1)

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d) rank = 2

determinant is singular

Not one to one or onto

nullity is

x+2Y =0 AND Y+z =0

(2, -1, 1) is the nullity