Chapter 1: Linear Equations in Linear Algebra
State two types of a consistent system of linear equations
- 1 solution - infinitely many solutions
State 4 scenarios which result in linear dependence
- a vector is the zero vector - if vectors are scalar multiples of each other - free variables - cols > row
State 2 relationships with regards to linear algebra transformations and pivots
- one-to-one ↔ pivot in every column - onto ↔ pivot in every row
Describe the term 'linear equation'
An equation that can be written in the form a₁x₁ + a₂x₂ + ... + aₙxₙ = b, where b and the co-efficients a₁,...,aₙ are real or complex numbers
Is the matrix [-4 3] a vector?
No
Define the system of linear equations
a collection of one or more linear equations involving the same variables
Define pivot position
a location in a given matrix A that corresponds to a leading 1 in the reduced echleon form of A
With regards to solutions, define a inconsistent system of linear equations
a system with no solutions
Describe the concept of a vector
a vector is a matrix with only one column
State the relationship between free variables and non-trivial solutions to a homogous system
free variables → non-trivial solutions
State the relationship between non-trivial solutions to a homogous system and columns of a A
non-trivial solutions -> columns are linearly dependent
State a criterion for 2 equivalent linear systems
the same solution set
Define span
the set of all possible linear combinations of a given list of vectors
Define solution set
the set of all possible solutions
State a property of a linear algebra vector transformation when A is a square matrix
this function is either injective and surjective, or neither
