RECORD. RECORD A Vector Space: the Definition.

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Presentation transcript:

RECORD

A Vector Space: the Definition

Exercises: Does the following set together with the given operations represent a vector space?

Exercises: Does the following set together with the given operations represent a vector space?

Exercises: Does the following set together with the given operations represent a vector space?

COLLABORATE: Exercises: Does the following set together with the given operations represent a vector space?

COLLABORATE: Exercises: Does the following set together with the given operations represent a vector space?

Exercises: Does the following set together with the given operations represent a vector space?

Properties of Vector Spaces :

Subspaces of Vector Spaces: Check to see if there are any properties inherited from V:

Subspace Criterion in two forms:

More Examples: The set of all symmetric matrices. The set of all lower triangular matrices. The set of all upper triangular matrices. The set of all diagonal matrices.

Example: Subset of the Vector Space That is Not a Subspace

Example: Lines and Planes Through the Origin as Subspaces of R2 and R3

The Intersection of Subspaces

The Span of the Set of Vectors

Example: Spaces Spanned by One or Two Vectors in 3-space Span of a subset in V is a subspace of V:

Exercises: Determine if the following are subspaces of Mnxn.