Rank-Nullity Theorem: In linear algebra, the rank-nullity theorem states that for any linear transformation from a finite-dimensional vector space V to itse...
Rank-Nullity Theorem:
In linear algebra, the rank-nullity theorem states that for any linear transformation from a finite-dimensional vector space V to itself, the following are equivalent:
Rank(T) = dim(V)
Null(T) = V
Here's a formal proof of the theorem:
Proof:
1. If Rank(T) = dim(V), then Null(T) = 0.
Since dim(V) is the dimension of the vector space, any linearly independent subset of V will span the entire space.
Therefore, Null(T) = {0} since the only linearly independent subset of V with 0 elements is the empty set.
2. Conversely, if Null(T) = V, then Rank(T) = dim(V).
We need to show that the rank of T is equal to the dimension of the vector space.
Since Null(T) = V, the rank of T is the number of linearly independent vectors in the set of vectors in V.
This is precisely equal to the dimension of V, proving the desired claim.
Examples:
Rank(T) = 2, V = R^3 (here T is the standard projection onto the plane in R^3)
Rank(T) = 1, V = {0}
Rank(T) = 0, V = R^4
The rank-nullity theorem provides a powerful connection between linear transformations and vector spaces. It allows us to determine the dimension of a vector space and identify all possible linear transformations that preserve the dimension of the space