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

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Presentation transcript:

RECORD

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.

Linear independence: Examples:

Establishing dependence/independence Exercises:

Revisit the example from page 3:

Results about dependent and independent sets:

Exercises:

Geometric Interpretation of Linear Dependence: Sets of Two Vectors

Geometric Interpretation of Linear Dependence: Sets of Three Vectors

Basis for the Vector Space: Definition

Exercises:

Exercises:

Dimension of the Vector Space:

Coordinate Representation of Vectors: