The rank-nullity theorem
Let be a linear transformation. Then there is a theorem that:
RankT + nullity T = n
This theorem is said to be equivalent to:
dim(Ran(T))+ dim(Kernel(T))= n
[n is the dimension of domain]
So here's my question: how can we use the first one to derive/prove the second?
This is my unsuccessful attempt:
And for nullity(T):
nullity(T) = dim(null(T))
Since null(T) = ker(T)
So, nullity(T) = dim(ker(T))
rank(T) = dim(row(A))
Since row(A) = null(A) and null(A) = ker(T)
dim(row(A)) = dim(ker(T))
I will be great if anyone could show me how to prove this. Thanks.
By definition, for a linear transformation T, dim(Range(T)) = Rank(T). Whats your definition?
Originally Posted by Roam
You seem to be confusing a linear transformation with its matrix(after choosing a basis).
P.S: What is A?