Math 215.01, Spring Term 1, 2021 Final Exam Information and Review
The final exam for Math 215.01, Spring Term 1, 2021 is open book and open note, including problem sets and writing assignments. The learning goals for the class will be reflected in our exam: good writing and correct use of definitions, quantifiers, and mathematical logic. The problems on the exam will be much like our problem sets and writing assignments, but we will not repeat exactly the problems we have already worked in class, in our book, or in our homework.
The final exam is Tuesday, March 23, 9 a.m. — 12 p.m. (CDT). (We change to Daylight Savings Time on Sunday, March 14.) The exam will be released on Pweb a few minutes before 9. The exam is due at 12:30 p.m. (CDT). The additional 30 minutes is for you to scan and upload your exam to Pweb.
You will write the exam by hand, unless you are more comfortable typing. You are NOT expected to work the exam and then typeset the exam in the three-hour time period.
It is crucial to know the vocabulary for the class; you do not want to use time on the final looking up definitions which you need to work problems. I have listed vocabulary and their associated ideas below to help you review for the exam. Notice that the main result from Chapter 5 is the statement, not the proof, of the Rank-Nullity Theorem.
• sets of numbers: N, N+, Q, Z, R
• even integers and odd integers
• quantifiers
– the role of “there exists” in a definition versus “choose/fix” when using a “there exists” definition in a proof
– the role of “for all” in a definition versus “arbitrary” when proving a “for all” statement in a proof
– how to prove a “for all” statement
– how to prove a “there exists” statement
• the difference between an official definition of a term versus a proposition which gives a recharacterization of that term
• two sets A and B are equal
– using the technique of double containment to show two sets are equal
• the difference between “∈” and “⊆”
• injective function f : A → B
– using the definition of injective for linear transformations T : V → W, where V
and W are finite-dimensional vector spaces • surjective function f : A → B
– using the definition of surjective for linear transformations T : V → W , where V and W are finite-dimensional vector spaces
• compositionoftwofunctionsf :A→Bandg:B→C
• contrapositive of an if-then statement
• logically equivalent mathematical statements: the role of “if and only if”
– dividing an “if and only if” statement into two “if-then” statements for an “if and only if” proof
• systems of linear equations in m equations in n variables
– A consistent system of linear equations in m equations in n variables
– An inconsistent system of linear equations in m equations in n variables
– Connecting systems of linear equations to spans of vectors in Rm
– Connecting systems of linear equations to augmented matrices
– Elementary row operations, echelon form, and reduced echelon form
– Leading variables and free variables
– Writing a solution set of a linear system in parametric form
– Connection among echelon forms, leading entries, and unique solutions of systems
– Connection among echelon forms, leading entries, and consistent systems
– Remember that you will not have to carry out extensive row reduction operations on the exam.
• the zero vector (additive identity) of a vector space V • an additive inverse of a vector ⃗v in a vector space V • the vector space Rn, where n ∈ N+
• the vector space Pn, where n ∈ N+
• the vector space M2×2
2
• the sequence of vectors (u⃗1, u⃗2, . . . , u⃗n) is linearly independent
– linearly dependent sets of vectors
– the logical negation of linearly independent
– What can you say about a set which contains the zero vector of a vector space V?
– What can you say about a set of vectors which contains a linearly dependent subset?
• a linear combination of vectors {u⃗1, u⃗2, . . . , u⃗n} in a vector space V • the set of vectors {u⃗1, u⃗2, . . . , u⃗n} spans a vector space V
– a technical way to say that the set of vectors {u⃗1, u⃗2, . . . , u⃗n} does NOT span V (“there exists …”)
– If a set of vectors {u⃗1, u⃗2, . . . , u⃗n} spans a vector space V , does every nontrivial subset of {u⃗1,u⃗2,…,u⃗n} span V ?
• the technical definition: the set S ⊆ V is closed under addition (“for all…”)
• the technical definition: the set S ⊆ V is closed under scalar multiplication (“for all…”) • ThesetS⊆V isasubspaceofV.
– HowtoprovethatS⊆V isasubspaceofV
– HowtoprovethatS⊆V isNOTasubspaceofV – Connection between spans and subspaces
• T : V → W is a linear transformation.
– the technical definition: T : V → W preserves addition (“for all…”)
– the technical definition: T : V → W preserves scalar multiplication (“for all…”) – WhyisT(⃗0V)=⃗0W?
• the standard basis for Rn
• bijective linear transformation T : V → W
• In the following, T : V → W is a linear transformation, and V and W are vector spaces.
– range(T ) (distinguish between vectors in the domain and codomain) (use set no- tation)
– Connection between range and surjectivity 3
– Null(T ) (distinguish between vectors in the domain and codomain) (use set nota- tion)
– Connection between nullspace and injectivity
• the inverse of a bijective linear transformation T : V → W
– How do we define T−1(w⃗), where w ∈ W?
• a basis for a vector space V
• the dimension of a vector space V which has a finite spanning set
• Suppose V is an n-dimensional vector space and V has basis α. What is the linear transformation Coordα?
• the statement of the Rank-Nullity Theorem
• The standard matrix [T ] of a linear transformation T : R2 → R2
– The span of the columns of [T ]
• Suppose we have a linear transformation T : R2 → R2. What are the various forms that Null(T ) and range(T ) can take? How is this result connected to the Rank-Nullity Theorem?
• The role of matrix-vector multiplication in evaluating T(⃗v) when T : R2 → R2 is a linear transformation
• Suppose T : R2 → R2 is a linear transformation, and suppose α is a basis for R2codomain. How do you compute the matrix [T]α?
• What diagram is associated with [T], ⃗v, Coordα, [⃗v]α, and [T]α? Interpret the diagram.
• eigenvectors and eigenvalues of T : R2 → R2
• characteristic polynomial of [T ] (degree two only)
• a 2 × 2 diagonal matrix
• a 2 × 2 diagonalizable matrix
4