Lecture 2 and Lecture 3 (Feb 5)

Elementary row operations:

1. (Replacement) Replace one equation by the sum of itself and a multiple of another equation;
2. (Scaling) Multiply an equation by a nonzero number;
3. (Interchange) Swap two equations.

Echelon form: A m-by-n matrix is in echelon form if it satisfies

1. If a row is nonzero, then every row above it is also nonzero.
2. The leading entry in a nonzero row is in a column to the right of the leading entry in the row above. 
3. If a row is nonzero, then every entry below its leading entry in the same column is zero. 

Reduced Echelon form: A m-by-n matrix is in reduced echelon form if it satisfies

1. The matrix is in echelon form.
2. Each nonzero row has leading entry 1.
3. The leading 1 in each nonzero row is the only nonzero number in its column.

A pivot position in a matrix is the location containing a leading 1 in the reduced echelon form for the matrix.

A pivot column in a matrix is a column contining a pivot position.

Definitions: Basic variable and free variable.

Summary:

• The system has no solution if the last column is a pivot column of A.
• The system has infinitely many solutions if the last column is not a pivot column but there is at least 1 free variable.
• The system has 1 solution if there are no free variables, and the last column is not a pivot column.

Vector: a very special form of matrix where it is simply a column.

Definition of addition and scaler multiplication of vectors.

Some properties:

1. Associativity
2. Commutativity
3. Additive identity
4. Multiplication associativity
5. Inverse
6. Distributivity 1
7. Distributivity 2
8. Multiplicative identity