[SOLVED] Math 215.01, Spring Term 1, 2021 Final Exam Information and Review

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 :ABandg:BC
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 M22
2

the sequence of vectors (u1, u2, . . . , un) 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 {u1, u2, . . . , un} in a vector space V the set of vectors {u1, u2, . . . , un} spans a vector space V
a technical way to say that the set of vectors {u1, u2, . . . , un} does NOT span V (there exists )
If a set of vectors {u1, u2, . . . , un} spans a vector space V , does every nontrivial subset of {u1,u2,,un} 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) ThesetSV isasubspaceofV.
HowtoprovethatSV isasubspaceofV
HowtoprovethatSV 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 T1(w), 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 . What is the linear transformation Coord?
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 is a basis for R2codomain. How do you compute the matrix [T]?
What diagram is associated with [T], v, Coord, [v], and [T]? 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
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