Midterm Exam Topics

This a collection of things you should be comfortable with going into the midterm. It's not exhaustive, and I can pretty much guarantee that no question on the midterm will be a direct restatement of anything below, but if you understand everything in this document, you have the foundations to approach any question on the midterm.

Objects you should know

Vectors
- zero vector
- standard basis vectors
Matrices
- echelon forms
- reduced echelon forms
- zero matrix
- identity matrix
- 2D linear transformation matrices for:
  - rotation
  - dilation
  - contraction
  - reflection
  - shearing
  - projection
- 3D linear transformations for:
  - reflection
  - rotation
- matrix operations
  - multiplication
  - powers
  - inverse
  - transpose

Things you should know how to do

Systems of linear equations
- Given a system of linear equations:
  - determine if a given solution satisfies it
  - determine if it is consistent
  - find one of its solution, if one exists
  - find a general form solution which describes its solution set
  - write down its augmented matrix
  - write down its coefficient matrix
  - row reduce its augmented/coefficient matrix to echelon form
  - row reduce its augmented/coefficient matrix to reduced echelon form
  - determine if it has a unique solution
  - determine if it has infinitely many solutions
Vectors
- Given a vector equation:
  - determine if a given solution satsifies it
  - determine if it is consistent
  - find one of its solutions, if one exists
  - find a general form solution which describes its solution set
  - write down a system of linear equations which has the same solution set
- Given a set of vectors in ℝⁿ:
  - compute a given linear combination of them
  - find a vector which lies in their span
  - determine if a given vector lies in their span
  - find a vector which does not lie in their span, if one exists
  - determine if set spans all of ℝⁿ
  - determine if the set is linearly independent
  - determine a dependence relation for them, if one exists
  - determine if its span is the same as the span of another given set of vectors
- Given two vectors:
  - compute their inner product
  - find a linear equation which describes plane spanned by those vectors, given they are linearly independent
- Given one vector:
  - find two plane equations whose intersection is its span, given it is nonzero
  - draw it, given it is in ℝ²
Matrices
- Given a matrix equation:
  - determine if a given vector is one of its solutions
  - determine if it is consistent
  - find one of its solutions, if one exists
  - write down a system of linear equations which has the same solution set
  - write down a vector equation which has the same solution set
- Given a (m × n) matrix A:
  - determine if it is row equivalent to another given matrix
  - determine if it is in echelon form
  - determine if it is in reduced echelon form
  - determine its pivot positions
  - determine if the product Av for a given vector v is defined
  - compute the product Av for a given vector v, given it is defined
  - determine if the equation Ax = b has a solution for any choice of b
  - determine if the equation Ax = b has a unique solution for a given choice of b
  - determine if a given vector can be written as a linear combination of its columns
  - determine if its columns span ℝᵐ
  - determine if its columns are linearly independent
- Given a (2 × 2) matrix A:
  - draw the effect on the transformation on the unit square
- Given a (n × n) matrix A:
  - find its inverse
  - use the invertible matrix theorem to determine if it's invertible
Linear transformations
- Given a transformation:
  - identify its domain
  - identify its codomain
  - identify if its range is the same as its codomain
  - determine if it is linear
- Given a linear transformation:
  - determine its value on a linear combination of vectors
  - find the matrix implementing a linear transformation given:
    - the input+output behavior of the transformation on a set of vectors
    - a geometric picture/description in ℝ² or ℝ³ of how the transformation behaves
    - an algebraic description of the linear transformation
  - find a set of vectors which spans its range

Facts you should know how to use

every matrix has a unique reduced echelon form
a matrix is the augmented matrix of a inconsistent system if and only if any of its echelon forms have a pivot in the last column
two vectors are linearly dependent if and only if they are co-linear
the following are equivalent for a (m × n) matrix A:
- Ax = 0 has a unique solution
- the columns of A are linearly independent
- A has a pivot in every column
the following are equivalent for a (m × n) matrix A:
- Ax = b has a solution for any choice of b
- the columns of A span ℝᵐ
- A has a pivot in every row
for a (m × n) matrix:
- if m < n, then its columns are not linearly independent
- if n < m, then its columns do not span ℝᵐ
every condition of the invertible matrix theorem

Counterexample you should know

a consistent system of linear equations with more equations than unknowns
an inconsistent system of linear equations with more unknowns than equations
two distinct echelon forms that are row+equivalent, but have the same entries in every pivot position
two different sized sets of vectors which have the same span
a set of linearly dependent vectors in which one cannot be written as a linear combination of other others
a set of linearly dependent vectors such that every proper subset is linear independent
a linear transformation which changes the length of some but not all vectors
a linear transformation which changes the direction of some but not all vectors