Question

(iv). State Sylvester's law of nullity.​

Answer 1

Answer : The rank-nullity theorem states that the rank and the nullity (the dimension of the kernel) sum to the number of columns in a given matrix. If there is a matrix MM with xx rows and yy columns over a field, then

rank(M)+nullity(M)=y

This can be generalized further to linear maps: if T: V \rightarrow WT:V→W is a linear map, then

dim(im(T))+dim(ker(T))=dim(V)

The rank-nullity theorem is further generalized by consideration of the fundamental subspaces and the fundamental theorem of linear algebra.

The rank-nullity theorem is useful in calculating either one by calculating the other instead, which is useful as it is often much easier to find the rank than the nullity (or vice versa).

hope it'll help you :)