LINEAR ALGEBRA

ℝ

Set of all real numbers

ℝ⁺

Set of all positive real numbers

ℝ²

Set of all 2-dimensional (x, y) ordered pairs

ℝ³

Set of all 3-dimensional (x, y, z) ordered triples

ℤ

Set of all integers

ℤ⁺

Set of all positive integers

Linear Algebra

Studies vector spaces and transformations done to them

Vector

General term of all objects that can be added to and scaled

Inconsistent

System of equations with 0 solution

Consistent

System of equations with at least 1 solution

Parameter

A different variable name given that has a value that can be freely chosen

Free Variable

refers to the original variable that is replaced by a parameter

Leading Variable

refers to the original variable(s) that remain the same throughout an equation

Parametric Equation

used to describe lines in 3-dimensional spaces, planes, systems of equations with infinitely many solutions, etc…

Matrices

Used in linear algebra to store information

Augmented matrix

Matrix form of systems of equations

• the last column stores constants while prior columns hold coefficients

• heavily reliant on the organization of the systems of equations

Gaussian Elimination

solve by converting system into an augmented matrix and then said matrix into a simpler, more solvable form (reduced-row echelon form)

Elementary Row Operations

ways to manipulate an augmented matrix to achieve a desired result

Row Scaling

multiplying rows by nonzero constant (cR_i → R_i)

Row Swapping

Interchanging 2 rows; putting equation in different order [R_i ←→ R_i]

Row Equivalence

Adding a multiple of one row to another [cR_i + R_j → R_j]

Reduced Row Echelon Form

a matrix where:

1) any rows consisting of all 0s are at the bottom of the matrix

2) all non-zero rows have the entry of a 1 (leading 1)

3) leading 1s move further to the right with each row further down the matrix

4) columns containing leading 1 contains 0 everywhere else in the row, except for the constant

Row Echelon Form

a matrix where:

1) any rows consisting of all 0s are at the bottom of the matrix

2) all non-zero rows have the entry of a 1 (leading 1)

3) leading 1s move further to the right with each row further down the matrix

Homogenous System

a linear system in which all constants are equal to 0

Trivial Solution

a solution to homogenous systems where all variables = 0

Nontrivial Solution

a solution to homogenous systems in which a variable has a value

Solving with Gaussian Elimination

Using row operations:

• scaling to introduce leading 1 or simplify matrix

• equivalence to introduce 0s in columns with leading 1 established already

• swapping to make leading 1 appear further right