Linear Algebra - 1

Oct 24th 14-16pm Mid-term Exam

  • Vectors = Data
  • Matrices = Transformations

Chapter 1 : Linear Equations

Section 1.1 : Systems of Linear Equations

  • linear equation
  • system of linear equation
  • solution set
  • Equivalent : Two linear system have the same solution set

Solution:

  • no
  • exactly one
  • infinitely many

For a system of linear equation:

  • Consistent : one or infinite solutions
  • Inconsistent : no solution

Coefficient Matrix

image-20260902110349443

Augmented Matrix

image-20260902110408719

Matrix size : m×nm\times n matrix is a rectangular with mm rows and nn columns

Elementary Row Operations

  • Replacement : Row Addition
  • Interchange : Row Swap
  • Scaling : Row Scaling

Two matrices are called row equivalent if there is a sequence of elementary row operations that transforms one matrix into another one

If the augmented matrices of two linear systems are row equivalent, then the two systems have the same solution set.

Section 1.2: Row Reduction and Echelon Forms

Echelon form (or row echelon) : /ˈeʃəlɒn/

  1. All nonzero rows are above any rows of all zeros
  2. Each leading entry of a row is in a column to the right of the leading entry of the row above it
  3. All entries in a column below a leading entry are zeros

Reduced echelon form (or reduced row echelon form):

  1. The leading entry in each nonzero row is 1
  2. Each leading 1 is the only nonzero entry in its column

Theorem : Uniqueness of the Reduced Echelon Form

Each matrix is row equivalent to one and only one reduced echelon matrix.

Pivot

Get it from the reduced echelon form.

image-20260902114143268

  • Pivot Position : a location in A that corresponds to a leading 1 in the reduced echelon form of A
  • Pivot column

Row Reduction Algorithm

  • forward phase
  • backward phase

Pivot columns are called basic variables, while other are called free variables

Parametric Descriptions of Solution Sets:

image-20260902115036875