Midterm Exam Prep Material
Practice Exams
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
- Vectors
- 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
- Given a system of linear equations:
- 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
- Given a vector equation:
- 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
- Given a matrix equation:
- 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
- Given a transformation:
- Systems of linear equations
- 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