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In linear algebra, a subspace is a subset of a vector space that is itself also a vector space. Subspaces must satisfy the properties of being closed under addition and scalar multiplication.

For our exercise, consider two specific subspaces extracted from vector space \(V\): \( \operatorname{rad}(V) \) and its orthogonal complement \( \operatorname{rad}(V)^{\perp-} \). By working with these subspaces, we can explore how they combine to form the entire vector space \( V \) through a direct sum.

A direct sum, denoted as \( R \oplus S \), means that every vector in the entire set can be uniquely expressed as a combination of a vector from subspace \( R \) and a vector from subspace \( S \). This is what we aim to show鈥攖hat the vector space \( V \) can be decomposed into a direct sum of \( \operatorname{rad}(V) \) and \( \operatorname{rad}(V)^{\perp-} \). This ensures that each vector in \( V \) has a single, unique representation as a combination of vectors from these two subspaces.

An orthogonal complement functions as a vital component in separating a vector space into mutually exclusive subspaces. Consider vector space \( V \) and a subspace \( R \). The orthogonal complement of \( R \), denoted as \( R^{\perp} \), includes every vector in \( V \) that is orthogonal to \( every \) vector in \( R \).

For our particular context of vector space \( V \), we are dealing with \( \operatorname{rad}(V)^{\perp-} \), which is the orthogonal complement of \( \operatorname{rad}(V) \). Understanding orthogonal complements allows us to comprehend how vectors can be separated based on orthogonality.

Orthogonality ensures that vectors in \( \operatorname{rad}(V) \) do not overlap with those in \( \operatorname{rad}(V)^{\perp-} \) except at the zero vector level. This property is essential because it underpins the ability to decompose \( V \) into non-overlapping parts, facilitating unique representation and clear vector analysis.

The concept of unique representation is pivotal in distinguishing the direct sum decomposition of vector spaces. A vector space \( V \) can be divided into \( \operatorname{rad}(V) \) and \( \operatorname{rad}(V)^{\perp-} \) such that each vector \( v \in V \) can be written as \( v = r + s \), where \( r \in \operatorname{rad}(V) \) and \( s \in \operatorname{rad}(V)^{\perp-} \).

This decomposition leads to a unique representation. Having a unique representation means that there is only one way to express a vector as a sum of vectors from \( \operatorname{rad}(V) \) and \( \operatorname{rad}(V)^{\perp-} \).

To ensure uniqueness, suppose two different representations exist: \( v = r_1 + s_1 \) and \( v = r_2 + s_2 \). This would lead to \( r_1 - r_2 = s_2 - s_1 \). Since \( \operatorname{rad}(V) \) and \( \operatorname{rad}(V)^{\perp-} \) intersect at only the zero vector, it follows that both differences must be zero. Hence, \( r_1 = r_2 \) and \( s_1 = s_2 \), confirming that the representation is indeed unique.

Understanding this uniqueness is crucial for effectively working with vector spaces, facilitating precise computations and analyses.