Section 4.1: Vector Spaces and Subspaces

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• Section 4.1 Vector Spaces and Subspaces

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REVIEW
Recall the following algebraic properties of
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Definition A vector space is a nonempty set V of
objects, called vectors, on which are defined two
operations, called addition and multiplication by
scalars, subject to the ten axioms
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Examples
the set of polynomials of degree at most n
the set of all real-valued functions defined on
a set D.
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• Definition
• A subspace of a vector space V is a subset H of V
  that satisfies
• The zero vector of V is in H.
• H is closed under vector addition.
• H is closed under multiplication by scalars.

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Properties a-c guarantee that a subspace H of V
is itself a vector space. Why? a, b, and c in
the defn are precisely axioms 1, 4, and 6.
Axioms 2, 3, 7-10 are true in H because they
apply to all elements in V, including those in H.
Axiom 5 is also true by c. Thus every
subspace is a vector space and conversely, every
vector space is a subspace (or itself or possibly
something larger). -
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Examples

• The zero space 0, consisting of only the zero
  vector in V
• is a subspace of V.

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5. Given and in a vector space V,
let Show that H is a subspace of V.
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Theorem 1 If are in a vector
space V, then Span
is a subspace of V.
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Example