Lec 10 - RREF and Rank

Wednesday, January 25, 2012

2:16 AM

Recall:

A system of linear equations is homogeneous if the RHS contains only zeros.

Eg

The only solution of

We know

Thus the set is linearly independent.

Eg.

Solve

The solution set has vector equation

Theorem:

Rank

Definition

The rank of a matrix is the number of leading ones in the RREF of the matrix.

Eg.

Theorem

Proof:

Eg.

First, is the set linear independent?

Proof

Note that every homogeneous system is consistent.

If there is an infinite number of solutions, let

…

Be the homogeneous system.

Apply the Subspace Test:

The solution set is non-empty since the system is consistent.

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