Let V be a vector space. Prove that the intersection ∩Wi of any collection (finite or infinite)

Step 1: u000ADefine the intersection of subspaces. Let $u005C{W_iu005C}_{i u005Cin I}$ be a collection of subspa

Let V be a vector space. Prove that the intersection ∩Wi of...

Key Concepts

Intersection of Subspaces

The intersection of subspaces refers to the set of all elements that are common to all the subspaces in a given collection. This concept is important because it highlights that the intersection of subspaces can itself form a subspace, as any element in the intersection will satisfy the properties required by all of the intersecting subspaces.

Closure Properties in Proof

The method of proving the intersection of subspaces is itself a subspace relies on checking the closure properties: it must be shown that the intersection contains the zero vector, is closed under addition, and is closed under scalar multiplication. Since each of these properties is satisfied by every subspace in the collection, they are inherited by the intersection, thereby forming a valid subspace.

Vector Space

A vector space is an algebraic structure consisting of a set equipped with two operations—vector addition and scalar multiplication—that satisfy a specific set of axioms such as associativity, commutativity, distributivity, identity, and invertibility. These axioms provide a framework that allows for the study of linear combinations and the overall structure in which vectors operate.

Subspace

A subspace is a subset of a vector space that is itself a vector space under the same operations. To qualify as a subspace, the subset must contain the zero vector and be closed under both vector addition and scalar multiplication. This means that any combination of its elements, using the vector space operations, will also belong to the subspace.

Let $V$ be a vector space. Prove that the intersection $\cap W_i$ of any collection (finite or infinite) of subspaces $W_i \subset V$ is a subspace.

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