Rank-Nullity Theorem
How do we determine the dimension of the image and kernel given the dimension of a vector space?
julianna_burke: May 4, 2015, 9:58 p.m.
dim of the image(A) is the rank(A) (leading 1s) dim of the kernel (A) is the # of columns with out a leading one
henry_lee: May 4, 2015, 9:59 p.m.
The rank nullity theorem states that the dimensions of the kernel and the dimensions of the image together should add up to the dimensions of the transformation. In #2 on the practice midterm, for example, the image's rank is 3, meaning the dimensions of it (dim(Im)) is 3. Since the transformation's dimensions (dim(V)) is 4, then we can use the rank nullity theorem to find that the kernel's dimension (dim(ker))is 1.
julianna_burke: May 4, 2015, 10:02 p.m.
I believe the theorem goes something like: n (# of columns) = rank(A) + nullity (A) // dim of image is the rank and dimension of kernel is the nullity // so if rank = n (number of columns) then ker(A)=0