Exam proofs and definitions

Question 1

Define what a dimension is, what it means for a vector V to be infinite dimensional and what it means for a vector V to be finite dimensional

Answer 1

Definition 3.1.1. Let V be a vector space. If V has a basis consisting of n vectors, we
say that V has dimension n. The subspace {0} of V is said to have dimension 0. V is
said to be finite dimensional if there is a finite set of vectors that spans V ; otherwise,
we say that V is infinite dimensional. The dimesnion of V is denoted dimV .

Question 2

Define a coordinate vector

Answer 2

Let V be a vector space over a field F and let S = {v1,v2, . . . ,vn} be
an ordered basis of V . If v is any element of V , then v can be written in the form
v = c1v1 +c2v2 +. . .+cnvn where ci 2 F, i = 1, 2, . . . ,n. Thus, we can associate with
each vector v a unique vector c = (c1,c2, . . . ,cn)T 2 Fn. The vector c defined in this
way is called the coordinate vector of v with respect to the ordered basis S and is
denoted [v]S . The ci ’s are called the coordinates of v relative to S.

Question 3

Define what injective means

Answer 3

Definition 3.4.1. Let M : S -> S’ be a map. We say that M is injective if whenever
x, y 2 S and x ≠ y, then M(x) ≠ M(y). In other words, M is injective means that M
takes on distinct values at distinct elements of S. Another way of saying the same
thing, we can say that M is injective if and only if, given x, y ∈ S.

Question 4

Define a surjective

Answer 4

Definition 3.4.2. Let M : S -> S’ be a map. We say that M is surjective if the image of
M is all of S’, i.e., M(S) = S’

Question 5

Define the identity mapping of a set

Answer 5

If S is any set, the identity mapping IS is defined to be the map such
that I_s (x) = x for all x ∈ S.

Question 6

Define bijective

Answer 6

Definition 3.4.3. Let M : S -> S’ be a map. We say that M is bijective if M is both
injective and surjective

Question 7

Define an inverse map

Answer 7

LetM : S-> S0 be a map. We say that M has an inverse (or is invert-
ible) if there exist a mapping N : S’ -> S such that

(Look at def 4.4.5 for the rest)

Question 8

Define a translation

Answer 8

LetV be a vector space, u ∈ V be a fixed element in V and Tu : V ->V
be a map such that Tu(v) = v+u. We call Tu the translation by u.
If S is any subset of V , then Tu(S) is called the translation of S by u.

Question 9

Define an isomorphism

Answer 9

Let V and V’ be vector spaces over a field F. The map L : V ->V’ is
an isomorphism if it is linear and has an inverse, i.e., invertible.

Question 10

Prove the following:

Let V be a finite-dimensional vector space over a field F and let n =
dimV . Then V is isomorphic to Fn

Answer 10

Look at theorem 4.5.1