For each of the following matrices write down the rank and t

For each of the following matrices, write down the rank and the dimensions of N(R), C(R), N(R^T),  and C(R^T): R = (0 0 0 0), R = (1 0 0 1), R = (0 1 -1 0 17 0 0 0 1 20 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0), R = (1 0 0 0 1 0 0 0 1 0 0 0 0 0 0), R = (1 0 0 0 0 1 0 0 0 0 1 0)

Solution

a)

By rank nullity theorem

rank+nullity=2

rank=0 as 0 matrix

rank=0 =dim(C(R))

dim(N(R))=nullity=2

R^T=0 matrix and of size 2x2

Hence,

dim(N(R^T))=2

dim(C(R^T))=0

b)

R=R^T= Identity matrix

dim(N(R))=dim(N(R^T))=0

dim(C(R))=dim(C(R^T))=2

c)

Only two linearly independent columns

Hence, rank(R)=dim col(R)=2

R is of size: 5x5

So by rank nullity theorem

rank+nullity=5

HEnce, dim(N(R))=3

rank(A)=rank(A^T)=dim(C(R^T))

R^T is also of size 5x5

Hence dim(N(R^T))=3

d)

three linearly independent columns

Hence, rank(R)=3=dim(C(R))

rank+nullity=3

Hence nullity =dim(N(R))=0

dim(C(R^T))=dim(C(R))=3

rank(R^T)+nullity(R^T)=5

Hence, nullity(R^T)=dim(N(R^T))=2

e)

three linearly independent columns

Hence, rank(R)=3=dim(C(R))

rank+nullity=4

Hence nullity =dim(N(R))=1

dim(C(R^T))=dim(C(R))=3

rank(R^T)+nullity(R^T)=3

Hence, nullity(R^T)=dim(N(R^T))=0