midterm

Question 1

VECTOR SPACE

Answer 1

• ∃0∈V : 0+u=u, ∀u∈V
• u+v=v+u, ∀u,v∈V
• u(v+w) = (u+v)+w, , ∀u,v,w∈V
• ∃z∈V : u+z=0, ∀u∈V
• 1.u=u, ∀u∈V
• a.b.u = ab.u, ∀u∈V, ∀a,b∈F
• a.(u+v) = (a.u)+(a.v), ∀u,v∈V, ∀a∈F
• u.(a+b) = (a.u)+(b.u), ∀u∈V, ∀a,b∈F

Question 2

U is subspace of V (def.)

Answer 2

if U is subset of V that is a vector space with same vector addition and scalar multiplication as in V

Question 3

U⊆V subpace of V
⟺

Answer 3

cu+v ∈U whenever u,v ∈U and c ∈F

Question 4

null(A)

Answer 4

{x ∈F^n: Ax=0} ⊆F^n
columns in kernel

Question 5

col(A)

Answer 5

{Ax: x∈F^n} ⊆F^n
set of all linear combinations (span) of the columns of A

Question 6

list vs set

Answer 6

β = a1, a2,…, ar vs β = {a1, a2,…, ar}

Question 7

span(U)

Answer 7

set of all linear combinations of elements of U (may be set U⊆F^n or list β=U)

Question 8

span(∅)
span(u∈U)

Answer 8

= {0}
={cu: c∈F}

Question 9

PROPERTIES OF SPAN (thm.1.4.9,10)

Answer 9

1. span(U⊆V)⊆V
2. U ⊆ span U
3. U = span U ⟺ U⊆V
4. span(spanU) = span U
5. U ⊆W ⟹ span U ⊆ span W

Question 10

U⊆V is spanning set of V
β:=list of vectors in V is spanning list of V

Answer 10

span(U) = V
span(β) = V

Question 11

span(U ∪ W),
U, W⊆V

Answer 11

= (U + W) ⊆ V

Question 12

DIRECT SUM

Answer 12

U,W ⊆ V: U ∩ W = ∅
⟹
U⊕W bijective (every element unique)

Question 13

v1,…,vr LINEARLY DEPENDENT

Answer 13

∃c1,…,cr ∈F not all zero:
c1v1+…+crvr=0

Question 14

v∈V linearly dependent ⟺

Answer 14

v=0

Question 15

any list of vectors containing the zero vector and/or a repeated vector is linearly dependent

Question 16

β = v1,…,vr linearly dependent ⟹

Answer 16

v1,…,vr, v linearly dependent, ∀v∈V

Question 17

β = v1,…,vr linearly independent

Answer 17

c1v1+…+crvr=0 ⟺ c1=…=cr=0

Question 18

v1,…,vr linearly independent
then a1v1+…+arvr = b1v1+…+brvr
⟺

Answer 18

ai=bi, ∀i=1,…,r

Question 19

omitted vj from list β, r>=2

Answer 19

v1,…,vj-hat,…,vr
* linearly indep if β is

Question 20

β linearly indep. and does not span V ⟹

Answer 20

β, v linearly independent, ∀v∉β

Question 21

β linearly indep. and span(β)=V, then c1v1+…+crvr=0 nontrivial ⟹

Answer 21

v1,…,vj-hat,…,vr spans V, cj≠0

Question 22

BASIS of vector space V

Answer 22

β linearly independent: span(β)=V

Question 23

A = [a1 … an] ∈ Mn(F) invertible ⟹

Answer 23

a1,…,an is basis for F^n

Question 24

REPLACEMENT LEMMA

Answer 24

• β = u1,…,ur spans V≠∅
• v= Σciui ≠ 0

⟹

1. ∃j: cj ≠ 0
2. cj ≠ 0 ⟹ v, u1,…, uj-hat,…, ur spans V
3. cj ≠ 0 and β basis for V
   ⟹
   v, u1,…, uj-hat,…, ur is basis for V
4. • β basis for V, r>=2
   • v∉span{u1,…,uk}, k∈{1,2,…,r}
     ⟹
     ∃j∈{k+1,k+2,…,r}: v, u1,…, uk, uk+1, uj-hat,…, ur is basis for V

Question 25

βr basis for V, γn linearly independent ⟹

Answer 25

⟹ n<=r * n=r ⟹ γ basis for V

Question 26

βr, γn bases for V⟹

Answer 26

n=r

Question 27

DIMENSION V

Answer 27

- {v1,...,vn} is basis of V ⟹ dimV=n - V=∅ ⟹ dimV=0

Question 28

DIMENSION MATRIX

Answer 28

- A = [a1 ... an] ∈ F^(mxn) - β=a1,...,an ⟹ dim span β = dim col A =: rank A

Question 29

span(v1,...,vr)=V ⟹

Answer 29

- dimV=n<=r - ∃i1,...,in ∈ {1,...,r}: {vi1,...,vin} basis for V

Question 30

v1,...,vr linearly independent, dimV=n>r ⟹

Answer 30

∃w1,...,wn-r∈V: v1,..., vr, w1,..., wn-r basis for V

Question 31

β = v1,...,vn, dimV=n ⟹

Answer 31

1. β spans V ⟹ β basis for V 2. β linearly independent ⟹ β basis for V

Question 32

U subspace of V, dimV=n ⟹

Answer 32

- dimU<=n - dimU=n ⟺ U=V

Question 33

U, W subspaces of V, dimV<∞ ⟹

Answer 33

dim(U ∩ W) + dim (U + W) = dim U + dim W

Question 34

U, W subspaces of V, dimV<∞, k>0 ⟹

Answer 34

1. dimU + dimV > dim V ⟹ ∃v∈(U ∩ W): v≠0 2. dimU + dimV >= dim V + k ⟹ U ∩ W contains k linearly independent vectors

Question 35

A, B, C ∈ Mn(F): AB=I=BC ⟹

Answer 35

A=C

Question 36

A, B ∈ Mn(F) ⟹

Answer 36

AB=I ⟺ BA=I

Question 37

β-BASIS REPRESENTATION FUNCTION

Answer 37

- the function [.]_β :V->Fn defined by [u]_β = [c1...cn]^T where β = v1,...,vn is basis for finite-dim. V and u∈V is any vector written as (unique) linear comb. u= c1v1+...+cnvn - c1,...,cn are "coordinates of u" w.r.t. basis β. - [u]_β is "β-coordinate vector of u"

Question 38

LINEAR TRANSFORMATION T:V->W

Answer 38

T(cu+v) = cT(u) +T(v) ∀c∈F,∀u,v∈V

Question 39

LINEAR OPERATOR T:V->W

Answer 39

V=W

Question 40

SET OF LINEAR TRANSFORMATIONS SET OF LINEAR OPERATORS

Answer 40

L(V,W) L(V)

Question 41

LINEAR TRANSFORMATION INDUCED BY A

Answer 41

T_A: F^(n) → F^(n): x↦Ax, A∈Mmxn(F)

Question 42

KerT T:V->W

Answer 42

= {v∈V|T(v) = 0}

Question 43

RanT T:V->W

Answer 43

= {w∈W|∃v ∈V : T(v)=w}

Question 44

T∈L(V,W) is one-to-one ⟺

Answer 44

ker(T) = {0}

Question 45

LINEAR TRANSFORMATION PROPERTIES

Answer 45

- T(cv) = cT(v) - T(0) = 0 - T(-v) = -T(v) - T(a1v1+...+anvn) = a1T(v1)+...+anT(vn)

Question 46

- β = v1,..., vn basis for V, T∈L(V,W) - v = c1v1+...cnvn ⟹

Answer 46

- Tv = c1Tv1+...cnTvn - RanT = span(Tv1,...,Tvn} - dimRanT <= n

Question 47

β-γ change of basis matrix

Answer 47

γ[I]β = [[v1]γ ... [vn]γ] (describes how to represent each vector in basis β as linear combination of vectors in the basis γ)

Question 48

inverse of γ[I]β

Answer 48

β[I]γ = [[w1] β ... [wn] β]

Question 49

- β = v1,..., vn basis for V, dimV=n>0 - S∈Mn(F) invertible ⟹

Answer 49

∃γ basis for V: S=β[I]γ

Question 50

β = a1,..., an basis for F^(n) ⟹

Answer 50

A = [a1...an] ∈ Mn(F) invertible

Question 51

A∈Mn(F) invertible ⟺

Answer 51

rankA=n

Question 52

- dimV=n>0 - β, γ bases for V - S= γ[I]β ⟹

Answer 52

- S invertible - γ[T]γ = S β[T]β S^(-1)

Question 53

- dimV=n>0 - S invertible - β basis for V ⟹

Answer 53

∃γ basis for V: γ[T]γ = S β[T]β S^(-1)

Question 54

A, B∈Mn(F) "similar over F"

Answer 54

∃S∈Mn(F): A=SBS^(-1)

Question 55

A,B∈Mn(F) similar ⟺

Answer 55

A=β[T]β and B= γ[T]γ, where β, γ bases for some V: dimV=n

Question 56

∃λ∈F: (A-λI) similar to (B-λI) ⟹

Answer 56

A similar to B

Question 57

A,B∈Mn(F) similar ⟹

Answer 57

- A-λI similar to B-λI, ∀λ∈F - TrA=TrB - detA=detB

Question 58

Similarity is equivalence relation

Answer 58

- reflexive - symmetric - transitive

Question 59

DIMENSION THEOREM FOR LINEAR TRANSFORMATIONS

Answer 59

dim ker T + dim ran T = dim V, T∈L(V,W)

Question 60

dimV=dimW, T∈L(V,W) ⟹

Answer 60

kerT=∅ ⟺ ranT=W

Question 61

DIMENSION THEOREM FOR MATRICES

Answer 61

- dim null A + dim col A = n - m=n ⟹ [nullA= ∅ ⟺colA=Fn] A ∈Mmxn(F)

Question 62

INNER PRODUCT on V

Answer 62

function <.,.> : VxV -> F satisfying ∀u,v,w∈V and ∀c∈F: - real >=0 - = 0 ⟺ v=0 - = + - = c - = __

Question 63

INNER PRODUCT SPACE

Answer 63

vector space V endowed with innerproduct

Question 64

standard inner product on F^n

Answer 64

= v*u = Σui(vi)_

Question 65

standard inner product on Pn

Answer 65

= ∫p(t)(q(t))_dt

Question 66

standard inner product on Mn(F)

Answer 66

= tr(B*A) = Σaij(bij)_

Question 67

standard inner product on C(F, [a,b])

Answer 67

= ∫(a,b)p(t)(q(t))_dt *when[a,b] = [-π,π], divide integral by π

Question 68

ORTHOGONAL u,v∈V

Answer 68

= 0 u⊥v

Question 69

ORTHOGONAL SUBSETS A,B ⊆V

Answer 69

every u∈A, v∈B u⊥v

Question 70

ORTHOGONAL PROPERTIES

Answer 70

- u⊥v ⟺ v⊥u - 0⊥u, ∀u∈V - v⊥u, ∀u∈V ⟹ v=0

Question 71

= , ∀u∈V ⟹

Answer 71

v=w

Question 72

NORM DERIVED FROM INNER PRODUCT

Answer 72

||v|| = √ referred to as norm on V

Question 73

Euclidean norm

Answer 73

||v||2 = √ = √(Σ|vi|^2)

Question 74

Frobenius norm

Answer 74

||A||2 = √ = tr(A*A) = Σ|aij|^2

Question 75

L^2 norm

Answer 75

||f|| = √( ∫(a,b)|f(t)|^2dt)

Question 76

DERIVED NORM PROPERTIES

Answer 76

1. "nonegativity" ||u|| real >=0 2. "positivity" ||u||= 0 ⟺ u=0 3. "homogeneity" ||cu|| = |c|||u|| 4. "pythagorean theorem" =0 ⟹ ||u+v||^2 = ||u||^2 + ||v||^2 5. "parallelogram identity" ||u+v||^2 + ||u-v||^2 = 2||u||^2 + 2||v||^2

Question 77

UNIT VECTOR u

Answer 77

||u|| = 1

Question 78

NORMALISATION of u

Answer 78

u/||u||

Question 79

CAUCHY SCHWARZ INEQUALITY

Answer 79

* || <= ||u||||v|| * || = ||u||||v|| ⟺ u,v linearly dependent i.e. one is scalar multiple of other

Question 80

TRIANGLE INEQUALITY FOR DERIVED NORM

Answer 80

* ||u+v||<= ||u||+||v|| * ||u+v||= ||u||+||v|| ⟺ one is real non-neg. scalar multiple of other

Question 81

POLARISATION IDENTITIES (4.5.24)

Answer 81

u,v∈V * F=R ⟹ = 1/4 (||u+v||^2 - ||u-v||^2) * F=C ⟹ = 1/4 (||u+v||^2 - ||u-v||^2 + i||u+iv||^2 - i||u-iv||^2)

Question 82

NORM on V

Answer 82

function ||.|| : V -> [0, ∞) with following properties for ∀u,v∈V and ∀c∈F: - ||u|| real >=0 - ||u||=0 ⟺ u=0 - ||cu|| = |c|||u|| - ||u+v||<= ||u||+||v||

Question 83

l1 norm

Answer 83

||u||1 = |u1| + ... + |un|

Question 84

l∞ norm

Answer 84

||u||∞ = max{|ui|: 1<=i<=n}

Question 85

Euclidean norm

Answer 85

||u||2 = √ (|u1|^2 + ... + |un|^2)

Question 86

UNIT BALL of normed space V

Answer 86

{v∈V: ||v||<=1}