Definitions And Basic Rules

Question 1

Vector Space

Answer 1

A set V equipped with the following data:
D1: Addition operation (sum of 2 vectors im V is also in V)
D2: contains zero vector
D3: scalar multiplication (scalarXv must be in V)

Question 2

8 vector space rules

Answer 2

1. v+w = w+v (COMMUTATIVE)
2. (v+w)+u = v+(w+u) (ASSOCIATIVE)
3. 0+v = v+0 = v (ZERO VECTOR IS ADDITIVE IDENTITY)
4. k.(v+w) = kv+kw, k is a scalar (DISTRIBUTIVE)
5. (k+l).v = kv+lv, k and l are scalars (DISTRIBUTIVE)
6. k.(l.v) = (kl).v , k, l scalars (ASSOCIATIVE)
7. 1.v = v (1 IS MULTIPLICATIVE IDENTITY)
8. 0.v = 0 (0 scalar X any vector = 0 vector)

Question 3

Additive inverse

Answer 3

(-v) := (-1)v

Question 4

Subspace

Answer 4

A subspace that satisfies all the vector space rules (see vector space card)

Question 5

Degree of a polynomial

Answer 5

The highest power of x

Question 6

Degree of a trig polynomial

Answer 6

The highest multiple of x in the formula

Question 7

Linear combination

Answer 7

A linear combination of a finite collection v1,v2,…,vn of vectors in a vector space V is a vector of the form a1v1+a2v2+…+anvn where a1, a2, …, an are scalars

Question 8

Span

Answer 8

A list of vectors spans a vector space if every vector in the vector space can be written as a linear combination of the vectors in the list

Question 9

Linearly independent

Answer 9

v1, …, vn is linearly independent if a1v1+…+anvn =0 has only the trivial solution a1=a2…=an=0

Question 10

list is linearly independent iff …

Answer 10

all vectors are non-zero and cannot be expressed as linear combinations of the others

Question 11

Linear combination of preceding vectors proposition

Answer 11

v1, v2, …, vn is a list of vectors in V. The following are equivalent:

1. The list is linearly dependent
2. Either v1 = 0, or for some r in {2, 3, …n} vr is a linear combination the v1, v2, …, v(r-1) vectors

Question 12

Bumping off proposition

Answer 12

Suppose l1, l2, …, lm is a linearly independent list of vectors in V and s1, s2, …, sn spans V. Then m<= n

Question 13

Basis

Answer 13

A list of vectors that is linearly independent and spans V

Question 14

Finite-dimensional

Answer 14

A vector space is finite dimensional if it has a basis e1, e2, …, en (you can count the basis vectors)

Question 15

Dimension of V

Answer 15

din V:= the number of vectors in a basis for V

Question 16

Invariance of dimension theorem

Answer 16

If e1, e2, …, en and f1, f2, …, fm are bases of a vector space V, then m=n

Question 17

Bases give coordinates proposition

Answer 17

A list of vectors e1, e2, …, en is a basis for V iff every vector in V can be written as a linear combination v = a1e1+a2e2+…+anen in precisely one way

Question 18

Sifting algorithm

Answer 18

1. Start with a list of vectors v1, v2, …,vn in a vector space V
2. Consider each vector vi consecutively. If vi=0, or if it is a linear combination of the preceding vectors, remove it
3. At the end, the resulting list will be linearly independent, by the linear combination of preceding vectors proposition

Question 19

Coordinate vector of v with respect to the basis B

Answer 19

v = a1b1=a2b2+…+anbn

v := [a1, a2, …, an]^T in Col(n)

Question 20

Change of basis matrix from B to C

Answer 20

B and C are both bases for V. The change of basis matrix from B to C is the matrix whose columns are the coordinate vectors of the B basis with respect to the C basis
P(C

Question 21

Change of basis theorem

Answer 21

B and C are bases of V.

v = P(C

Question 22

Linear map

Answer 22

A function T: V –> W satisfying:
T(v+v’) = T(v) + T(v’)
T(kv) = kT(v)
where V, v’ are vectors and k is scalar

Question 23

Sufficient to define a linear map on a basis proposition

Answer 23

Suppose B = {e1, e2, …, en} is a basis of V and let W be a vector space with w1, w2, …, wn in W. There exists a linear map T:V–>W such that ei |–> wi , i = 1, …, n

Question 24

T after S

Answer 24

The composite of T with S.

If S: U –> V and T: V–> W then the composition of T after S is the function T(S(u))

Question 25

Isomorphism

Answer 25

A linear map T: V–>W is an isomorphism of vector spaces if there exists a linear map S:W–>V such that
T after S = id(v) and S after T = id(w)
We say that S is the inverse of T

Question 26

Uniqueness of inverse

Answer 26

If T:V–>W is a linear map and S,S’:W–>V are inverses of T, then S=S’

Question 27

Isomorphic

Answer 27

Two vector spaces V and W are isomorphic if there exists and isomorphism T:V–>W

Question 28

Theorem: V and W are isomorphic iff…

Answer 28

dimV=dimW

Question 29

.

Answer 29

V –> Col(n)
v |–> v
is an isomorphism

Question 30

The matrix of T with respect to the bases B and C

Answer 30

The matrix whose columns are the coordinate vectors of T(bi) with respect to the basis C
[T](C

Question 31

T(v) = …

Answer 31

[T](C

Question 32

Functoriality of a matrix on a linear map theorem

Answer 32

Let S:U–>V and T:V–>W be linear maps between finite dimensional vector spaces. Let B, C, D be bases for U, V and W respectively.
[T after S](D

Question 33

Corollary of Functoriality of a matrix on a linear map theorem

Answer 33

T is invertible iff [T](C

Question 34

Kernel

Answer 34

T:V –> W
Ker(t) := {v in V : T(v) = the zero vector in W}
(null space)

Question 35

Image

Answer 35

T:V –> W
Im(T) := {w in W : w=T(v) for some v in V}
(image/ column space)

Question 36

Nullity

Answer 36

Nullity(T) := Dim(Ker(T))

Question 37

Rank

Answer 37

Rank(T) := Dim(Im(T))

Question 38

Rank-nullity theorem

Answer 38

Nullity(T)+Rank(T) = Dim(T)

Question 39

Injective

Answer 39

f(x) = f(x’) ==> x = x’

Question 40

Surjective

Answer 40

For every y in Y, there is an x in X such that f(x) = y

Question 41

T:V –> W then T injective iff

Answer 41

Ker(T) = {the zero vector in v}

Question 42

T:V –> V where V is finite dimensional

T injective iff

Answer 42

T surjective

Question 43

T: V–>W is an isomorphism iff

Answer 43

T is injective and surjective

Question 44

Eigenvalue

Answer 44

T: V–>V

l is an eigenvalue of T if there exists a non-zero vector such that T(v) = Lv

Question 45

l is an eigenvalue of T iff… (3 statements)

Answer 45

T-Lid is not injective
T-Lid is not surjective
T-Lid is not an isomorphism

Question 46

Eigenvector

Answer 46

A non-zero vector such that T(v)=Lv for some l in the Reals

Question 47

Determinant of a linear operator

Answer 47

det(T) := det([T}(B

Question 48

Characteristic polynomial

Answer 48

det(Lid-T)

Question 49

Eigenspace

Answer 49

E(L) := {v in V : T(v) = Lv} in V