Math 108B, W 21 Review Outline for Second Midterm
** Only material studied since the first midterm is listed on this review sheet, but you
should understand and be ready to use the earlier material in combination with this.
Material: Sections 7A (skip 7.16), 7B (7.23-24), 7C (7.37-43), 8A, 8B (8.20-30), 8C.
Reminder: This midterm will take place on Monday 22 February from 5–6 PM (Pacific
Standard Time).
Definitions.
conjugate transpose of a matrix
orthonormally diagonalizable operator
normal operator
isometry
generalized eigenvector of an operator
generalized eigenspace of an operator
nilpotent operator
strictly upper triangular matrix
multiplicity of an eigenvalue
block diagonal matrix
characteristic polynomial
monic polynomial
minimal polynomial
one polynomial divides another
one polynomial is a multiple of another
Symbols and Abbreviations.
T ∗
TFAE
G(λ, T )
Theorems. This is a list of the major theorems (and corollaries, etc.) that we have
developed, with phrases labelling them. You don’t need to memorize the numbers of these
theorems, but you should know the statements. (The phrases listed here are not full
statements.) The numbers are listed here to help you find things in the book.
Note: Some of these results concern arbitrary vector spaces and some concern arbitrary
inner product spaces. However, certain results require the vector space or inner product
space to be finite dimensional. If “finite dimensional” is in the hypothesis of a result, it
cannot be applied to arbitrary vector spaces. Some results require the field to be C, and
some require the field to be R. Any of these kinds of hypotheses must be included when
you give a complete statement of a result.
Existence of adjoints (discussion under 7.2)
Null spaces and ranges of adjoints (7.7)
The matrix of an adjoint (7.10)
Self-adjoint T with Tv ⊥ v for all v implies T = 0 (7.14 & 7.16)
Normality versus norm conditions (7.20)
Normal operators and their adjoints have the same eigenvectors (7.21)
Eigenvalues for a normal operator and its adjoint are conjugates (in class)
Orthogonality of eigenvectors of a normal operator (7.22)
Normal operators and adjoints restricted to invariant subspaces (in class)
Complex Spectral Theorem (7.24)
Characterizations of isometries (7.42)
Isometries are normal (in class)
Description of isometries over C (7.43)
Null spaces of powers of operators (8.2 & 8.3)
Limit to growth of null spaces (8.4)
Direct sum of null space and range for dimV -th power of an operator (8.5)
Null space description of generalized eigenspaces (8.11)
Linear independence of generalized eigenvectors (8.13)
Generalized eigenspaces make a direct sum (in class)
The dimV -th power of a nilpotent operator is zero (8.18)
Matrices of nilpotent operators (8.19)
Invariance of null spaces and ranges of polynomials in an operator (8.20)
Description of operators on complex vector spaces (8.21)
Bases of generalized eigenvectors (8.23)
Sum of multiplicities of eigenvalues of an operator (8.26)
Block diagonal matrices for operators (8.29)
Degrees and roots of characteristic polynomials (8.36)
Cayley-Hamilton Theorem (8.37)
Existence and uniqueness of minimal polynomial (8.40)
Minimal polynomial of T divides q iff q(T ) = 0 (8.46)
Minimal polynomial divides characteristic polynomial (8.48)
Roots of minimal polynomial are eigenvalues (8.49)
Vocabulary, Grammar, Reading, Writing. Remember that mathematics is a lan-
guage as well as a subject. The vocabulary consists of words and symbols with precise
definitions. Some of these you already have in your background; the new ones appearing
in this course are listed under “Definitions” and “Symbols” above. The basic grammar of
mathematics is built up from logic and the expressions (both verbal and symbolic) used
to make mathematical statements. Vocabulary and grammar are both needed in order to
read or write a piece of mathematics.
All of these aspects of mathematical language will be tested. Questions of the following
types may appear:
Give a precise definition of the concept or symbol “· · · ”.
Give a precise statement of the theorem, lemma, proposition, or corollary “· · ·”.
Such a statement should include the complete hypotheses and the full conclu-
sion(s).
Some proofs will also be asked for – proofs of statements done in the book and/or in
class; proofs that appeared as homework problems; or proofs of new statements.
Finally, there may be questions involving small calculations.

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