MA2001 cheatsheet PDF

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cRi , c  =0 {(1 + s, 2s, s)∣s ∈ R}

{

Ri ↔ Rj Ri + cRj , c ∈ R

x=t  = 2t − 1

x1 , x2 , … , xn = 0

Ri

(A T )T = A B=C

E 3 E 2 E1 A = B 1 1 A = E1 1 E2 E3 B

(A + B) T = AT + B T A

B2 m× B1 = B 2

AB1 = AB2

C1

C2 m× C1 = C2

C1 A = C2 A

a A=[ c A 1 =

(AB)T

A=[

b ] d

1 d [ ad  bc c

(cA)T = cAT

c

B1

b ] a

T

=B A

a c

T

b ] d

A

 det(A) = ad  bc  =0

A



ifE isanelementarymatrixofthesamesizeasA, det(B) = det(E) det(A) = det(EA) kR



AB R ↔R m

A, B

c

R +kRm

A  B

(cA)1 = 1c A 1

cA A

A  B 1

T 1

T

(A )

AB

(AB)1 = B 1 A1

det(B) =  det(A)



det(B) = det(A)

A, B det(cA) = A = 0

(A  )1 = (A1 )

det(A)

det(AB) = det(A) det(B )

AB = I

A, B

 c

det(A1 ) =

B

1 det(A)

B1 = A

BA = I ifAisaninvertiblematrix,then 1 A 1 = adj (A) det(A)

A, B A AB

AB A

det(A) = det(AT )

BA B

i = A

det(Ai ) b

det(A) = 0

det(Ai ) det(A)

ih

det(E ) = k



;

= (A )

(A1 )1 = A

A 1 = B;

;

1 T

A1

A

det(B) = k det(A)

c

;

det(E) = det(E) = 1

S

R

V S

R

S ++ = 0 thegeneralsolutiontothelinearsystem {  −  + 2 = 1

LetV beasubsetofR .V isasubspaceofR if V = span(S)forsomevectors1 , 2 , … , k ∈ R  .

V V

V

(,  , ) = ( 12 − 32 , − 12 + 12 , )where ∈ R

(ii)(Closedunderlinearcombinations)∀,  ∈ V , ,  ∈ R,  +  ∈ V

{(,  , ) ∣  +  +  = 0and −  + 2 =

V

S

thecoordinatevectorofV relativetoS , () = (c1 , c2 , … , ck ) ∈ Rk

 1,  2, … ,  k

[]  V

 ∈ V ⊆ R R

V

V V

V W

V

dim(V ),

W ⊆

W



 1,  2, … ,  k V

()

V

R W

S V

{( 12 − 32 , −21 + 12 , ) ∣  ∈ R}

V = R

(i)(Containstheorigin)O ∈ V

∅

1}

V

AsubspaceV ⊆ R

 =

()  ∈ Rk

k

dim(R  ) = 

E = {e 1 , e2 , … , e } e1 = (1, 0, … , 0), e2 = (0, 1, … , 0), e = (0, 0, … , 1) LetS = {1 , 2 , ⋯ ,  } ⊆ R . 0 ∈ span(S ) ∀ 1 , 2 , … ,  ∈ span(S)

R

V

c1 , c2 , … , c ∈ R,

c 1 1 + c2 2 + ⋯ + c  ∈ span(S )

()E = (1 , 2 , … ,  ) = 

⎡ ⎢ ⎢ ⎢ ⎣

S

S

V

∣S∣ = k

S 1 , 2 , … ,  ∈ V

c 1, c 2, … , c  ∈ R

(c1 1 +c 2 2 + ⋯ + c  ) S = c1 (1 )S + c2 (2 )S + ⋯ + c ( )S

S S

V

V

∣S∣ = k

S

,  ∈ V ,  =   () S = () S

S

V S

 letS = {1 , 2 , … ,  } a(S) = R  thelinearsystem 1 k1  2 k2   isconsistent∀k1 , k2 , … , k ∈ R ⋮ ⋮   k 

k

S

V

V



∣S∣ = k

REFhasnozerorow ⇒ a(S) = R V 

1 , 2 , … ,  (1 )S , (2 )S , … , ( )S Rk

LetS = {1 , 2 , ⋯ , k } ⊆ R  . k <  ⇒ a(S)  = R

LetU beasubspaceofvectorspaceV . Then dim(U ) ≤ dim(V ). If dim(U ) = dim(V )thenU = V .



R2 R

Rk

3

T

span{1 , 2 , … ,  } = V  span{(1 )S , (2 )S , … , ( )S } = Rk

∣T ∣ > dim(V )

V

span{1 , 2 , 3 } ⊆ span{1 , 2 }  1 , 2

 1 , 2 ,  3

P = [ [1 ]T [ 2] T {1 , 2 , … , k }

A

[ 1 2 ∣  1 ∣ 2 ∣  3 ] span{1 , 2 , 3 } ⊆ V

[ T

A = 0

1 , 2

V

if 1 , 2 , … ,   ∈ a(S) ⇒ a{1 , 2 , … ,  } ⊆ a(S )

P P −1

det(A)  =0

A=B

A

A ⊆ B∧B ⊆ A

c 1 1 + c2 2 + ⋯ + ck k = 0 S = {0} S S

R A



R

(∗) k 1 , 2 , … , k−1 , a{1 , 2 , … , k−1 } = a{1 , 2 , … , k−1 , k }

∣

⋯

[k ]T ]

G-JElimination

S ]  [ I []T = P []S S

T T

S

S=

∣

P ]

R

A

= span{1 , 2 ,  ,  }  R = columnspaceofAT ⎡ 1 ⎤

A

whereA = ⎢   , i = [ai1 ⎣   orA = [c1



A A = 0

ai2



ai ] , rank(A) = min{, }

⎡ a 1 ⎤ c ] , c =⎢   ⎣ a   R

A = span{c1 , c2 ,  , c }  R = rowspaceofAT

nullity(A)  dim(R ) = 

A

  det(A)  =0

rank(A T ) + nullity(AT ) = numberofrowsinA rank(A) + nullity(A) = numberofcolumnsinA

rank(0) = 0,

rank(I ) = ,

rank(A)  min{, }

rank(A) = AT



A

rank(AB)  min{rank(A), rank(B )}

= {A    R  } A = b b

A

 A

(A  b ) A = b



A 

{0} A = b

= { +   isanelementofthenullspaceofA}  ✓

A

nullity(A) = dim(nullspaceofA)

d(u, v) = u  v  u =

u  v = 0,

uu =

u21

+

u22

++

u2n

=

 2

Rn

0

u  v = uv T = ∑ni=1 ui vi = u1 v1 + u 2v 2 +  + un vn u  = cos 1 (



v

uv uv )

=

u2 +v 2 uv 2 cos 1( 2uv 

⇏

)

2 ++un vn ) Rn :  = cos1 ( u1 v1 +u2uv v

u  v2 = u2 + v 2  2uv cos 

E = e1 , e2 ,  , en 

u  Rn

Rn

uv =vu

S

(cu)  v = u  (cv) = c(u  v )

u

0 Rn

V V

w  (u + v) = w  u + w  v

V

V

S S = dim(V )

V

u1 , u2 ,  , uk 

foranyw  Rn , w  u1 w  u2 w  uk u1 + u2 +  + uk u1  u1 u2  u2 uk  uk istheprojectionofwontoV .

 span(S) = V

(u  v)  w  = u  (v  w ) cu = cu uu0 uu=0  u=0 u  v  uv  u + v  u + v 

LetS = u1 , u2 ,  , un beanorthogonalbasisforV . Thenforanyw  V , w  u2 w  uk w  u1 w= u1 + uk u2 +  + uk  uk u 2  u2 u1  u1 w  u1 w  u2 w  uk , ,, ) (w)S = ( u 1  u1 u 2  u 2 uk  uk

d(u, w)  d(u, v) + d(v, w) LetS = u1 , u2 ,  , un beanorthonormalbasisforV . Thenforanyw  V , w = (w  v1 )v1 + (w  v2 )v2 +  + (w  vk )vk (w)S = (w  v1 , w  v 2,  , w  vk )

Rn

V

v1 , v2 ,  , vk 

V foranyw  Rn , (w  u1 )u1 + (w  u2 )u2 +  + (w  uk )uk istheprojectionofwontoV .

u  Rn

Ax = b

 p = Au

b

 p = Au

b

A

Ax = b

u A

E = e 1, e 2

S = u1 , u2  x, y, x , y

a1 , a2 ,  , an A =

 b = Au [a1 a2  an ]

u1 = (c , in ) = e1 c  + e2 in  T

 u

T

u2 = ( in , c ) = e1 in  + e2 c 

A Ax = A b

S

b

E

c   in  ] P =[ in  c 

 Ax = p A

x p

PT AT Ax p

u d(u, p)  d(u, v)

E

T

=A b

xy

V

v = (x, y)  R2 ,

fallv  V

 b  Au  b  Av 

A

fallv  Rn

e1 , e2 , u1 , u2

b AT Ax = A T b

u

S

x y  (v)S = (x  , y  )

x c  in  x [  ] = [ v ] S = P T [ v ]E = [ ][ ]  in  c  y y x = x c  + y in  y = x in  + y c 

Au u A A1 = AT

Rn

A A

S

T

P

S

T

P P T = P 1 =

T

S

Rn

A

n

E (A) 

E u  Rn

Au = u



u

A A



1 ⎡ 1

 A

Au = u

u

Au  spanu u, v, w [ u v w ] 1A[ u v w ] =

Rn u

A

m

⎡ 1 ⎢ 0 m A = P⎢ ⎢  ⎣ 0

P P A

1

A A

det(I  A) = 0 det(I  A)

0 1 2

 

 0

 

0 m 2

 

 0

 

0 ⎤ 0  1 P   1 n 

0 ⎤ 0   Pm   m  n

AP

P

 u  Rn \0  (I  A)u = 0  det(I  A) = 0

0 = P⎢ ⎢ ⎢  ⎣ 0

i

m  Z+

E u 



1

(I  A)x = 0

[ u v w ]



i  =0

A

A

nn

A A

A

n

P T AP = D D

P

 P

A

AT = A

A A

det(I  A) = (  1 )r 1 (   2)r2  (   k ) rk here 1 ,  , k aredistincteigenaluesofA. Aisdiagonaliable  dim(E i ) = ri foreacheigenaluei   S i = ri

Ax = 0

r 1 + r2 + + r k = n det(A)  =0

ri Rn

A

Rn

A rank(A) = n 0

A

A

n

A

R  R T : R  R

: R   R =

T (0) = 0

R

T

⎛⎡ 1  ⎞ ⎡ a 11 a 12  a1  ⎡ 1  ⎜⎢ 2  ⎟ ⎢ a 21 a 22  a2  ⎢ 2  T ⎜⎢  ⎟ = ⎢ ⎢  ⎜⎢   ⎟ ⎢   ⎢    ⎝  ⎠ a1 a2  a     ⎡ a11 1 a 12 2  a 1   ⎢ a21 1 a 22 2  a 2    = ⎢ ⎢     a1 1 a 2 2  a   for( 1,  2,  ,   )T  R

(aij )

T

1 , 2 ,  ,   R

c 1, c 2,  , c   R

T (c 1 1 + c2 2 +  + c  ) = c1 T (1 ) + c 2 T (2 ) +  + c T ( ) T R(T ) 

T :R  R

T

R(T ) = {T ()    R  }  R



A

[T (e1 ) T (e2 ) 

⎡ a 1j  ⎢ a 2j  = T (ei ) = Ae i = ⎢ ⎢    ai 

T (e )]

R(T ) = span{T (1 ), T (2 ),  , T ( )} R(T ) =

ih

A R(T )

T

A

rank(T ) = dim(R(T )) = dim(columnspaceofA) = rank(A)

R V

W T :V W



T (c + d ) = cT () + dT ()

,   V

c, d  R

let{1 , 2 ,  ,  }beabasisforR . foranyvector  R , = c1 1 + c2 2 +  + c  forsomec1 ,  , c  R {1 , 2 ,  ,  } T ()  T (1 ), T (2 ),  , T ( )

I : R  R I

T ker(T ) R

I

ker(T ) = {  T () = 0}  R

R

I

O : R  R O

0

R

T S (T  S )() = T (S ())

R

R

  R

  R (T  S )() = T (S ()) = T (A) = BA BA

ker(T ) = T

A ker(T )

nullity(T ) = dim(ker(T )) = nullity(A)

T S rank(T ) + nullity(T ) =  = rank(A) + nullity(A) = numberofcolumnsinA...