MATH 2270 MIDTERM 3 CHECKLIST
1. Overview
The exam covers 4.4-4.7, 5.1-5.5, and 6.1-6.3. There will be 8 questions, each
worth 5 points. You will be asked to do 5, and the maximum possible score is 25.
The exam will be in our usual room, B4 510, at the usual time of 0800-0850 on
Friday 13 April.
2. What to bring
You will need a writing implement of some kind. Paper will be provided.
3. Homework and the exam
Homework 8 is covered on the exam. Make sure you hand it in by 1700 on the
day of the exam.
4. Calculators
The exam is closed book and calculators are not allowed.
5. Topics List
•Statement and proof of the Unique Representation Theorem (Theorem 7, p.
246).
•Definition of and computations with PB, the change-of-coordinates matrix.
•Definition of isomorphism. Be able to identify when two finite-dimensional
vector spaces are isomorphic. To that end, you need to know the definition of
dimension in the context of a vector space.
•Statement and proof of theorems 9 and 10 in 4.5. Statement of theorems 11
and 12 in the same section.
•Definition of the rank of a matrix. Computations involving ranks and dimen-
sions of null spaces.
•Be able to change bases in Fn. (Re: Theorem 15, p. 273)
•Definition and computations with elementary matrices.
•Definition and computation of eigenvalues,eigenvectors, and eigenspaces.
This will likely involve the characteristic matrix,polynomial, and equation.
This includes both real and complex eigenvalues.
•Statement and use of Theorems 1 and 2 in Section 5.1.
•Definition of similar matrices.
•Statement and use of Theorem 5, p.320. Thus, you will need to know what it
means for a matrix to be diagonalizable.
•Statement and proof of Theorem 6, p.323.
•Computations involving different representations of the same linear transfor-
mation.
•Definition of inner product and norm in Rn(not Cn).
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