I am a little confused about dimensions, if the nullspace of a matrix is spanned by the 0 vector, does that mean the dimension of the nullspace of this matrix is 0? In the problems I attached, both A and B reduced to the identity matrix. Note (2) is supposed to be dim(N(B)) and dim(col(B)) instead of dim(N(A)) and dim(col(A)). Since they both reduce to I, that means a pivot is in every column, which means that the dimension of N(A) and N(B) is 0 right? Consequently, this means dim(Col(A)) and dim(col(B)) is 3 right?