Midterm Review
Geometric Algorithms
Boston University
Fall 2023
This a collection of things you should be comfortable with going into the midterm. It is 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. You will have to add to it as you see fit.
Objects you should know
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 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:
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
Counterexample you should know
- a consistent system of linear equations with more equations than unknowns
- an inconsistent linear 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