THEOREMS

Question 1

v(^)j

Answer 1

• omitting vj from list
• ## if list was linearly indep then so is ommited list

Question 2

• THEOREM 2.1.9
• COR. 2.1.10

Answer 2

• if β=u1,…,un basis for V and γ=v1,…,vr is linearly independent, then r<=n
  if r=n then γ basis for V
• if v1,…,vn and w1,…,wr are bases of V then r=n

Question 3

• COR. 2.2.8
• THEOREM 2.2.9

Answer 3

• Let V n-dim and β=v1,…,vn∈V
• if β spans V, then it is a basis
• if β linearly independent then it is basis
• U subspace of n-dim V. then U finite dim. with dimU<=n
  dimU=n ⟺U=V

Question 4

COR. 2.5.3

Answer 4

V,W finite dim. with dimV=dimW and T∈L(V,W)
then KerT={0} ⟺ ranT=W

Question 5

THEOREM 5.2.5

Answer 5

V finite dim. inner product space
β=u1,…,un orthonormal basis
v∈V
then v= Σ<v,ui>ui
||v||^2 = Σ|<v,ui>|^2
and basis representation of v w.r.t. β is [v]_β = [<v,u1> … <v,un>]^T

Question 6

COR. 5.3.12
COR 5.3.13

Answer 6

• every finite dim. inner product space has an orthonormal basis
• every orthonormal system in a finite dim. inner product space can be extended to an orthonormal basis

Question 7

THEOREM 7.1.8

Answer 7

if U finite dim. subsoace of innerproduct space, then (U^(⊥))^(⊥)=U

Question 8

THEOREM 7.3.9

Answer 8

Let X, Y∈Mn×r have orthonormal columns
Then col X = col Y ⟺ ∃U ∈ Mr unitary: X = YU.

Question 9

THEOREM 7.3.10

Answer 9

Let U be a finite-dimensional subspace of V

1. P_{0} = 0 and P_V = I
2. ran P_U = U
3. ker P_U = U^(⊥)
4. ​v − P_U(v) ∈ U^(⊥), ∀v ∈ V
5. ||P_U (v)|| ≤ ||v||, ∀v ∈ V
   (equality ⟺ v ∈ U)

Question 10

THEOREM 7.3.11

Answer 10

let V be finite dim., U subspace of V
1. (P_(u)^(⊥)) = I - (P_u)
2. (P_u)(P_(u)^(⊥))=(P_(u)^(⊥))(P_u)=0
3. (P_(u))^(2) = (P_u)
4. (P_u) = (P_(u))^*

Question 11

THEOREM 7.3.14

Answer 11

let V be finite dim., and P∈L(V)
then
P orthogonal projection onto ranP
⟺
P self adjoint and idempotent

Question 12

THEOREM 13.1.2

Answer 12

A∈Mn(F) positive semi-definite
⟺
A hermitian and all eigenvalues non-negative
⟺
∃B∈Mn(F) : A=BB
⟺
for some m>0, ∃B∈Mmxn(F): A=BB

Question 13

THEOREM 13.1.8

Answer 13

A∈Mn(F) positive definite
⟺
A hermitian and all eigenvalues positive
⟺
A is positive semi-definite and invertible
⟺
A is invertible and A^(-1) is positive definite
⟺
∃B∈Mn(F) invertible : A=BB
⟺
for some m>=n, ∃B∈Mmxn(F): rankB=n and A=BB

Question 14

THEOREM 13.1.9

Answer 14

• let A∈Mn(F) positive semi-definite
  then
  1. trA>=0 with strict equality ⟺ A≠0
  2. detA>=0 with strict equality ⟺ A positive definite

Question 15

THEOREM 13.1.10

Answer 15

suppose B∈Mmxn(F) and let A=BB
then
1. nullA=nullB
2. rankA=rankB
3. A=0 ⟺ B=0
4. colA = colB
5. if B normal then colA=colB

Question 16

cor. 13.1.12

Answer 16

let A∈Mn be positive semidefinite and partitioned as
A=[[A11 A12][A12*][A22]]
then A11 is positive semidefinite and
colA12 ⊆ col A11
consequently
rankA12<=rankA11
if A is positive definite then A11 is positive definite

Question 17

cor 13.1.18

Answer 17

let A∈Mn be positive definite and let x∈Cn
then <Ax,x>=0 ⟺ Ax=0

Question 18

theorem 13.1.20

Answer 18

let A,B∈Mn(F) and a,b∈R

1. if A,B hermitian then aA+bB hermitian
2. if a,b nonnegative and A,B are positive semidefinite, then aA+bB positive semidefinite
3. if a,b positive, A,B positive semidefinite, and at least one of A,B are positive definite then aA+bB positive definite

Question 19

theorem 13.1.23

Answer 19

let A,S∈Mn, B∈Mm, suppose A,B positive semidefinite

1. A+B positive semidefinite,
   (positive definite ⟺ A,B both positive definite)
2. S*AS positive semidefinite,
   (positive definite ⟺ A positive definite and S invertible)

Question 20

theorem 13.1.24

Answer 20

let A=[aij]∈Mn hermitian, suppose has nonnegative diagonal entries

1. if A diagonally dominant, then positive semidefinite
2. if A diagonally dominant and invertible, then positive definite
3. A strictly diagonally dominant, then positive definite
4. A diagonally dominant, has no zero entries, and |akk||>R’k(A) for at least one k∈{1,…,n} then A positive definite

Question 21

THEOREM 14.1.3

Answer 21

let A∈Mmxn(F) with rankA=r and q=min{m,n}
- δ1^2,…, δr^2 are positive eigenvalues of A
- Σδi^2 = trAA = trAA = (||A||F)^2
- A, A*, A^T and A() have the same singular values
- the singular values of cA are |c|δ1,…,|c|δq