Lec 3 - Basis & Higher Dimensions

Monday, January 09, 2012

9:24 AM

Q: How do we find the smallest spanning set?

Definition

A set of vectors is linearly dependent if one vector in the set is a linear combination of the remaining vectors.

Otherwise, the set of vectors is linearly independent.

Is there a nice mathematical expression to define linear dependence?

Better Definition

Proof:

Q.E.D.

Eg.

Consider the equation

We have 3 equations and 3 unknowns:

Sub into equation 1:

Our set is linearly dependent.

Note: In order for a spanning set to be as small as possible, it must be linearly independent.

Definition

Ex.

Eg.

Surfaces in Higher Dimensions

Definition

Subspaces

For a set of vectors to be closed under linear combinations, we must be able to apply the operations of vector addition and scalar multiplication.

Definition

Definition

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