Section 1.2 : Row Reduction and Echelon Forms

Topics

• Row reduction algorithm
• Pivots, and basic and free variables
• Echelon forms, existence and uniqueness

Echelon Form and RREF

A rectangular matrix is in echelon form if

1. All zero rows (if any are present) are at the bottom.
2. The first non-zero entry (or leading entry) of a row is to the right of any leading entries in the row above it (if any).
3. All elements below a leading entry (if any) are zero.

A matrix in echelon form is in row reduced echelon form (RREF) if

1. All leading entries, if any, are equal to 1.
2. Leading entries are the only nonzero entry in their respective column.

Note, it is not required for each column to have a leading term

Example of a matrix in echelon form

Fun fact: The zero matrix and identity is a matrix in RREF, recall notation for both zero and identity matrices

Pivot Position, Pivot Column

A pivot position in a matrix A is a location in A that corresponds to a
leading 1 in the reduced echelon form of A.

• Can be identified without needing to transform matrix to RREF

A pivot column is a column of A that contains a pivot position.

• If a column with a pivot indicates there is a single value for the corresponding variable (i.e., it is not a free variable)

Exercise: Express the matrix in row reduced echelon form and identify
the pivot columns.

→ Solution, matrix with pivot columns for , , and

Row Reduction Algorithm

Row equivalence: Matrices that can be converted between each other through elemetary row operations*

The algorithm we used in the previous example produces a matrix in RREF. Its steps can be stated as follows.

Step 1a. Swap the 1st row with a lower one so the leftmost nonzero entry is in the 1st row
Step 1b. Scale the 1st row so that its leading entry is equal to 1
Step 1c. Use row replacement so all entries below this 1 are 0
Step 2a. Swap the 2nd row with a lower one so that the leftmost nonzero entry below 1st row is in the 2nd row

etc. . . .
Now the matrix is in echelon form, with leading entries equal to 1.

Last step. Use row replacement so all entries above each leading entry are 0, starting from the right.

Basic and Free Variables

Consider the augmented matrix

The leading one’s are in first, third, and fifth columns. So:

• The pivot variables of the system are , , and .
• The free variables are and . Any choice of the free variables leads to a solution of the system.
• Upon arriving to the RREF of the system, the pivots/basic variables can be expressed in terms of the free variables (and then be used to parameterize the solution)

Note that does not have basic variables or free variables. Systems have variables.

• Meaning, basic and free variables only exist when solving the matrix to produce a vector
• A matrix by itself is an object with properties

By the way, I think this is neat

Good to keep in mind notation and what we are doing. We solve for , which represents a vector when multiplied by matrix A, yields vector .

• As discussed in the first lecture, there may be one, zero, or infinite number of solutions.

Existence and Uniqueness

Section 1.3 : Vector Equations

Topics

• Vectors in and their basic properties
• Linear combinations of vectors
  • Characterize their span
  • How a set of vectors are related to each other geometrically

Motivation

We want to think about the algebra in linear algebra (systems of equations and their solution sets) in terms of geometry (points, lines, planes, etc).

• This will give us better insight into the properties of systems of
  equations and their solution sets.
• To do this, we need to introduce n-dimensional space , and vectors inside it.

Vectors… how does this fit into ?

• The equation expresses whether or not the vector is within the span of the columns matrix A, given by the solution
• A matrix specifies a collection of vectors

Ordered n-tuples of real numbers ()

Recall that denotes the collection of all real numbers

Let be a positive whole number. We then define:
→ = all ordered -tuples of real numbers

Representations

• When n = 1, gemetrically, this is the number line
• When n = 2, we can think of as a plane
  • Every point in this plane can be represented by an ordered pair of real numbers, its xand ycoordinates
  • Exercise: Sketch the point (3, 2) and the vector
• As n increases by 1, we add an additional dimension to the “variable space”

Vectors

Vector algebra

Parallelogram rule for vector addition

Vectors have the following properties

Scalar multiple:

Vector addition:

→ Note that vectors in higher dimensions have the same properties!

Linear Combinations and Span

Visualization in interactive textbook

Geometric Interpretation of Linear Combinations

Example - Span

Solution
Construct an augmented matrix, use row reduction to arrive at RREF

→ The matrix is inconsistent, meaning the vector is not within the span

For this problem, the augmented matrix represents

• The variables c_1 and c_2 represent scalar weights used in linear combinations

The Span of Two Vectors in R^3

In the previous example, did we find that is in the span of and ?

In general: Any two non-parallel vectors in span a plan that passes through the origin. Any vector in that plane is also in the span of the two vectors.

Next lecture: 05.20 Lecture - MATH1554