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.

adjoint of an operator

conjugate transpose of a matrix

self-adjoint operator

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)

Linearity of adjoints (7.5)

Properties of adjoints (7.6)

Null spaces and ranges of adjoints (7.7)

The matrix of an adjoint (7.10)

Eigenvalues of self-adjoint operators (7.13)

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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