91Ó°ÊÓ

The Rank-Nullity Theorem is a fundamental concept in linear algebra that relates the dimensions of different subspaces associated with a linear transformation. For a linear transformation \( T : V \rightarrow W \), where \( V \) and \( W \) are vector spaces, the theorem states that:

\[ \text{dim}(V) = \text{rank}(T) + \text{nullity}(T) \]

Here, \( \text{rank}(T) \) refers to the dimension of the range of \( T \), while \( \text{nullity}(T) \) refers to the dimension of the kernel of \( T \).

- **Rank (\(T\))**: The rank of \( T \) is the number of linearly independent columns in the matrix representation of \( T \), or simply the dimension of the image of \( T \).
- **Nullity (\(T\))**: The nullity measures the "failure" of \( T \) to be injective; it is the dimension of the kernel of \( T \).

Using the Rank-Nullity Theorem in the given exercise, we see why \( \text{rank}(T) = \text{rank}(T^{2}) \) implies \( \text{nullity}(T) = \text{nullity}(T^{2}) \). Since the total dimension \( \text{dim}(V) \) is composed of both rank and nullity, keeping rank the same for both transformation \( T \) and its square \( T^2 \) necessitates the nullities to be identical.

A vector space is a collection of objects called vectors, which can be added together and multiplied (scaled) by numbers, known as scalars in the context of the field over which the vector space is defined. The basic idea of a vector space relies on the ability to perform these two operations while satisfying certain axioms, such as associativity, commutativity of vector addition, and distributivity.

Some key characteristics of a vector space include:

- **Closure**: The sum of two vectors in the space is also a vector within the same space, and scalar multiplication of a vector results in another vector in the space.
- **Zero Vector**: There exists a zero vector \( \mathbf{0} \) in the space such that adding it to any other vector \( v \) yields \( v \).
- **Inverse Elements**: For each vector \( v \), there exists an inverse \( -v \) such that \( v + (-v) = \mathbf{0} \).

In the context of the exercise, we're dealing with a finite-dimensional vector space \( V \), which means \( V \) has a finite basis, and all linear transformations including \( T \) involve vectors from this space.

When discussing linear transformations, two fundamental subspaces come into play: the kernel and the range.

- **Kernel**: The kernel of a linear transformation \( T : V \rightarrow W \) is the set of all vectors in \( V \) that \( T \) maps to the zero vector in \( W \). Formally, it's denoted as:
\[ \text{ker}(T) = \{ v \in V \mid T(v) = 0 \} \]
The kernel is crucial for determining the nullity of \( T \) and understanding how \( T \) might not be injective (one-to-one).

- **Range**: The range (or image) of \( T \) is the set of all vectors in \( W \) that can be written as \( T(v) \) for some vector \( v \) in \( V \). The range gives us the rank of \( T \), showing which vectors are "covered" by the transformation.
\[ \text{range}(T) = \{ T(v) \mid v \in V \} \]

In this exercise, proving that the intersection \( \text{range}(T) \cap \text{ker}(T) = \{0\} \) helps to determine the characteristics of non-singular transformations, highlighting that \( T \) has no overlap (except at the zero vector) between the space it "annihilates" and the space it actually "covers" with transformations.