Math 108B, W 21 Review Outline for First Midterm

Material: Sections 5C, 6A, 6B (skip 6.22, 6.37, 6.38), 6C

You can either print out the exam and write your answers in the spaces there, or write

your answers on blank paper. Be sure to write your full name and perm number on your

exam paper, and number your answers.

Reminder: This midterm will take place on Monday 25 January from 5–6 PM (Pacific

Standard Time).

Definitions.

eigenvalue

eigenvector

eigenspace

diagonal matrix

diagonalizable operator

dot product (in Rn or Cn)

inner product

Euclidean inner product

inner product space

norm

orthogonal vectors

orthogonal decomposition of one vector relative to another

orthonormal list of vectors

orthonormal basis

linear functional

orthogonal complement

projection onto one subspace along another

[This was only discussed in class, not in the book.]

orthogonal projection onto a subspace

Symbols and Abbreviations.

diag(a1, . . . , an)

E(λ, T )

x · y

〈u, v〉

‖v‖

u ⊥ v

ONB

〈−, u〉

U⊥

P

U

Theorems. This is a list of the major theorems (and corollaries, etc.) that we have

developed so far, 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, some 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.

Sum of eigenspaces (5.38)

Diagonalizability theorem (conditions for diagonalizability) (5.41)

Enough eigenvalues implies diagonalizability (5.44)

Basic properties of inner products (6.7)

Basic properties of norms (6.10)

Basic properties of orthogonality (6.12)

Pythagorean Theorem (6.13)

Orthogonal decomposition (6.14)

Cauchy-Schwarz Inequality (6.15)

Triangle Inequality (6.18)

Norm of an orthonormal linear combination (6.25)

Orthonormality implies linear independence (6.26)

Orthonormal lists of the right length are ONBs (6.28)

Expressing vectors as linear combinations of an ONB (6.30)

Gram-Schmidt Procedure (6.31)

Existence of ONBs (6.34)

Extending orthonormal lists to ONBs (6.35)

Riesz Representation Theorem (6.42)

Basic properties of orthogonal complements (6.46)

Direct sum of a subspace and its orthogonal complement (6.47)

Dimensions of orthogonal complements (6.50)

Orthogonal complements of orthogonal complements (6.51)

Properties of projections [in class]

Properties of orthogonal projections (6.55)

Minimizing distance to a subspace (6.56)

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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