Resumen-algebra 1 - tatata tata tatata tatatatata tata tatata tatatatata tata tatata tatatatata PDF

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tatata tata tatata tatatatata tata tatata tatatatata tata tatata tatatatata tata tatata tatatatata tata tatata tata...

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· ij

ij

mxn

1xn mx1

A =

n

(aij )ni·j=1 Pndiag (A) = (a11 , a22, ..., ann ) tr (A) = i=1 aii = a11 + a22 + ... + ann (a12, a2,3 , ..., an−1,n )

(a21, a3,2 , ..., an,n−1 )

At ∈ Mnxm(R)

A ∈ Mmxn (R)

→

t

→

t

A ∈ Mmxn (C) ∗ = a aij ji

A∗ ∈ Mnxm(C) a+ib = a−bi

A∗ = A

diag(d1 , d2 , ...dn )

n

mxn

[A + B ]ij = aij + bij λ

λ

[λA]ij = λaij

A, B, C ∈ Mmxn (R)

α, β ∈ (K)

(A + B) + C = A + (B + C) A+B =B +A A+0=A=0+A A + (−A) = 0 = −A + A

α(A + B) = αA + αB (α + β)A = αA + βA (αβ )A = α(βA) ∈ Mmxn (K)

[(A + B) + C]ij = [A + B]ij + cij = aij + bij + cij = aij + [B + C]ij = [A + (B + C )]ij [A + B ]ij = aij + bij = bij + aij = [B + A]ij [A + 0]ij = aij + 0 = aij = 0 + aij = [0 + A]ij [A + (−A)]ij = aij − aij = 0 = (−aij ) + aij = [−A + A]ij [α(A + B)]ij = α[A + B ]ij = α(aij + bij ) = αaij + αbij = [αA]ij + [αB ]ij = [αA + βB]ij [(α + β)A]ij = (α + β)aij = αaij + βaij = [αA]ij + [βA]ij = [αA + βA]ij [(αβ)A]ij = (αβ)aij = α(βaij ) = α[βA]ij = [α(βA)]ij [1A]ij = 1aij = aij = [A]ij



 [AB ]ij = ai1 ai2

 b1j n  X   b2j  aik bkj . . . ain   = ai1 b1j + ai2 b2j + . . . + ain bnj =   k=1 bnj

A ∈ Mmxn (K), B

C ∈ Mnxp(R)

D ∈ Mpxq (R), α ∈ (R)

(AB)D = A(BD) AIn = A A = AIm α(AB ) = (αA)B = A(αB )

Pn Pn Pn Pn Pn Pn Pn [(AB)D] = a b d = [ AB ] d j = ( a b )d j = ij ih hk kj ik k ih hk k k=1 aih bhk dkj = h=1 k=1 h=1 k=1 h=1 k=1 Pn Pn Pn (b d ) = a a [ BD ] = [A ( BD )] hk kj ih ih hj ij k=1 h=1 h=1 [AIn ]ij =

Pn

k=1

aik [In ]kj = aij

[In ]jj = 1

[In ]kj = 0

k 6= j

Pn P P [α(AB)]ij = α[AB ]ij = α( k=1 aik bkj ) = nk=1 αaik bkj = nk=1(αA)ik bkj = [(αA)B]ij Pn Pn Pn Pn [A(B + C)]ij = k=1 a k(B + C) = a (b + c ) = a b + a c = i kj ik kj kj ik kj ik kj k=1 aik bkj + k=1 k=1 Pn h=1 aih chj = [AB]ij + [AC]ij = [AB + AC]ij...