Lec 20 - Linear Independence & Basis

Friday, February 17, 2012

9:29 AM

Linear Independence

Definition

Eg

Solve the equation

We get the following homogeneous system of linear equations:

Row reduce the coefficient matrix:

Basis

What is the smallest spanning set of a vector space?

Theorem:

Proof:

Theorem

Proof:

Prove the contrapositive:

Both theorems together imply the smallest spanning set of any vector space is linearly independent.

Definition:

Eg

Eg

This gives a system of linear equations:

Row reduce the coefficient matrix:

Since the rank of coefficient matrix must equal to the number of columns, the homogeneous system has a unique solution (the trivial one).

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