Introdução à Álgebra Linear.pdf
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Introdução à Álgebra Linear.pdf


DisciplinaÁlgebra Linear I19.364 materiais281.685 seguidores
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A, temos que a aplicac¸a\u2dco linear
Km×n \u2212\u2192 L(V,W) que leva a matriz A na transformac¸a\u2dco linear AB \u2032B e´
a inversa da aplicac¸a\u2dco linear M : L(V,W) \u2212\u2192 Km×n que leva L em [L]B \u2032B.
Logo, M e´ um isomorfismo. \ufffd
Exemplo 1.7
Sejam A uma matriz m × n sobre o corpo K e L : Kn×1 \u2212\u2192 Km×1 a
transformac¸a\u2dco linear que leva X em AX.
Sejam B = {e1, . . . , en} e B \u2032 = {e1, . . . , em} as bases cano\u2c6nicas de Kn×1 e
Km×1, respectivamente.
Enta\u2dco, L(ej) = Aej = Aj e´ a j\u2212e´sima coluna da matriz A.
Como Aj =
m\u2211
i=1
Aijei, temos que A = [L]B \u2032B. \ufffd
J. Delgado - K. Frensel 76 Instituto de Matema´tica - UFF
Transformac¸a\u2dco Linear - noc¸o\u2dces ba´sicas
Proposic¸a\u2dco 1.9
Sejam V , W e Z espac¸os vetoriais sobre o corpo K de dimenso\u2dces n, m e
p, respectivamente.
Sejam T : V \u2212\u2192W e U : W \u2212\u2192 Z transformac¸o\u2dces lineares. Enta\u2dco,
[U \u25e6 T ]B \u2032\u2032B = [U]B \u2032\u2032B \u2032[T ]B \u2032B ,
onde B = {v1, . . . , vm}, B \u2032 = {w1, . . . , wm} e B \u2032\u2032 = {z1, . . . , zp} sa\u2dco bases de
V , W e Z respectivamente.
Prova.
Sejam A = [T ]B \u2032B, B = [U]B \u2032\u2032B \u2032 e C = [UT ]B \u2032\u2032B.
Enta\u2dco,
(U \u25e6 T)(vj) = U(T(vj)) = U
(
m\u2211
i=1
Aijwi
)
=
\u2211m
i=1AijU(wi)
=
m\u2211
i=1
Aij
p\u2211
k=1
Bkizk =
p\u2211
k=1
(
m\u2211
i=1
BkiAij
)
zk
=
p\u2211
k=1
(BA)kjzk .
Logo, Ckj = (BA)kj , \u2200k = 1, . . . , p e \u2200j = 1, . . . , n.
Ou seja, C = BA. \ufffd
Proposic¸a\u2dco 1.10
Sejam V e W espac¸os vetoriais de dimensa\u2dco finita. A transformac¸a\u2dco linear
L : V \u2212\u2192 W e´ um isomorfismo (ou seja, e´ invert\u131´vel) se, e somente se,
[L]B \u2032B e´ uma matriz invert\u131´vel, quaisquer que sejam as bases B de V e B \u2032
de W. Ale´m disso, se [L]B \u2032B e´ invert\u131´vel, enta\u2dco [L]\u22121B \u2032B = [L\u22121]BB \u2032.
Prova.
(=\u21d2) Seja T : W \u2212\u2192 V tal que T \u25e6 L = IV e L \u25e6 T = IW, isto e´, T = L\u22121.
Enta\u2dco,
[T ]BB \u2032[L]B \u2032B = [IV ]BB = I e [L]B \u2032B[T ]BB \u2032 = [IW]B \u2032B \u2032 = I,
ou seja, [L]B \u2032B e´ invert\u131´vel e [T ]BB \u2032 = [L\u22121]BB \u2032 e´ a sua inversa.
(\u21d0=) Sejam A = [L]B \u2032B e B = A\u22121. Seja T : W \u2212\u2192 V a transformac¸a\u2dco
linear BBB \u2032, ou seja, [T ]BB \u2032 = B.
J. Delgado - K. Frensel 77 Instituto de Matema´tica - UFF
Transformac¸a\u2dco Linear - noc¸o\u2dces ba´sicas
Enta\u2dco,
[T \u25e6 L]BB = [T ]BB \u2032[L]B \u2032B = BA = I.
Logo, (T \u25e6 L)(vj) = vj, \u2200j = 1, . . . , n, onde B = {v1, . . . , vn}.
Ou seja, T \u25e6 L = IV .
De modo ana´logo, temos que
[L \u25e6 T ]B \u2032B \u2032 = [L]B \u2032B[T ]BB \u2032 = AB = I.
Logo, (L \u25e6 T)(wi) = wi, \u2200i = 1, . . . ,m, onde B \u2032 = {w1, . . . , wm}.
Ou seja, L \u25e6 T = IW.
Obtemos, assim, que L e´ invert\u131´vel. \ufffd
Proposic¸a\u2dco 1.11
Sejam B1 e B \u20321 bases de V e B2 e B \u20322 bases de W, e L : V \u2212\u2192 W uma
transformac¸a\u2dco linear. Enta\u2dco,
[L]B \u20322B \u20321 = [IW]B \u20322B2 [L]B2B1 [IV ]B1B \u20321 .
Ou seja, o seguinte diagrama e´ comutativo:
VB1
L\u2212\u2212\u2212\u2212\u2192
[L]B2B1
WB2
[IV ]B1B\u20321
x\uf8e6\uf8e6IV IW\uf8e6\uf8e6y[IW ]B\u20322B2
VB \u20321
L\u2212\u2212\u2212\u2212\u2192
[L]B\u2032
2
B\u2032
1
WB \u20322
Prova.
Como [IW]B \u20322B2 [L]B2B1 = [IW \u25e6 L]B \u20322B1 = [L]B \u20322B1 , temos
[IW]B \u20322B2 [L]B2B1 [IV ]B1B \u20321 = [L]B \u20322B1 [IV ]B1B \u20321 = [L \u25e6 IV ]B \u20322B \u20321 = [L]B \u20322B \u20321 .
\ufffd
Note que ...
A matriz [IW ]B \u2032
2
B2 e´ s
matrizes de mudanc¸a de base
da base B2 para a base B \u20322 de
W. Analogamente, a matriz e
[IV ]B1B \u20321 muda a base B
\u2032
1
para a base B1 de V .
Observac¸a\u2dco 1.10
As matrizes [IW]B \u20322B2 e [IV ]B1B \u20321 sa\u2dco invert\u131´veis.
Notac¸a\u2dco
Se L : V \u2212\u2192 V e´ um operador sobre V e B e´ uma base de V ,
escrevemos
[L]B
def
= [L]BB .
J. Delgado - K. Frensel 78 Instituto de Matema´tica - UFF
Transformac¸a\u2dco Linear - noc¸o\u2dces ba´sicas
Em particular: se B e B \u2032 sa\u2dco bases de V e L : V \u2212\u2192 V e´ um operador
sobre V , enta\u2dco,
[L]B \u2032 = [I]B \u2032B[L]B[I]BB \u2032 .
Como, ale´m disso, [I]B \u2032B = ([I]BB \u2032)\u22121, temos que
[L]B \u2032 = P\u22121[L]BP ,
onde P = [I]BB \u2032, ou seja, se B \u2032 = {v \u20321, . . . , v \u2032n}, enta\u2dco Pj = [v \u2032j]B e´ a j\u2212e´sima
coluna de P.
Definic¸a\u2dco 1.10
Sejam A e B matrizes n× n sobre o corpo K.
Dizemos que B e´ semelhante a A sobre K se existe uma matriz n × n
invert\u131´vel P com entradas no corpo K, tal que
B = P\u22121AP .
Observac¸a\u2dco 1.11
Sejam B e B \u2032 bases de um espac¸o vetorial V sobre K e L : V \u2212\u2192 V um
operador linear. Enta\u2dco [L]B \u2032 e´ semelhante a [L]B.
Sejam, agora, A e B matrizes semelhantes n× n sobre o corpo K.
Sejam V um espac¸o vetorial de dimensa\u2dco n e B = {v1, . . . , vn} uma
base de V .
Consideremos o operador linear L : V \u2212\u2192 V tal que [L]B = A. Se
B = P\u22121AP, definimos os vetores
v \u2032j =
n\u2211
i=1
Pijvi , j = 1, . . . , n .
Afirmac¸a\u2dco: B \u2032 = {v \u20321, . . . , v \u2032n} e´ uma base de V . Note que ...
A matriz P e´ a matriz de
mudanc¸a da base B \u2032 para a
base B.De fato:
Seja U : V \u2212\u2192 V o operador linear tal que U(vj) = v \u2032j, j = 1, . . . , n.
Como [U]BB = P e P e´ invert\u131´vel, temos que U e´ invert\u131´vel. Logo, B \u2032
e´ uma base de V .
Ale´m disso, [I]BB \u2032 = P. Logo,
[L]B \u2032 = [I]B \u2032B[L]B[I]BB \u2032 = P\u22121[L]BP = P\u22121AP = B .
Isto e´, se A e B sa\u2dco semelhantes, existem bases B e B \u2032 do espac¸o
J. Delgado - K. Frensel 79 Instituto de Matema´tica - UFF
Transformac¸a\u2dco Linear - noc¸o\u2dces ba´sicas
V , tais que [L]B = A e [L]B \u2032 = B.
Exemplo 1.8
Seja T o operador linear sobre R3 definido por
T(x1, x2, x3) = (3x1 + x3,\u22122x1 + x2,\u2212x1 + 2x2 + 4x3) .
a. Determinar a matriz de T em relac¸a\u2dco a` base cano\u2c6nica B = {e1, e2, e3}
de R3.
Soluc¸a\u2dco: Como
T(e1) = T(1, 0, 0) = (3,\u22122,\u22121)
T(e2) = T(0, 1, 0) = (0, 1, 2)
T(e3) = T(0, 0, 1) = (1, 0, 4) ,
temos que,
[T ]B =
\uf8eb\uf8ec\uf8ed 3 0 1\u22122 1 0
\u22121 2 4
\uf8f6\uf8f7\uf8f8.
b. Determinar a matriz de T em relac¸a\u2dco a` base ordenada B \u2032 = {w1, w2, w3},
onde w1 = (1, 0, 1), w2 = (\u22121, 2, 1) e w3 = (2, 1, 1).
Soluc¸a\u2dco: Seja P = [I]BB \u2032 =
(
1 \u22121 2
0 2 1
1 1 1
)
.
Enta\u2dco,
[T ]B \u2032 = P\u22121[T ]BP.
Determinemos P\u22121.\uf8eb\uf8ec\uf8ed1 \u22121 20 2 1
1 1 1
\uf8f6\uf8f7\uf8f8 \u2190\u2192
\uf8eb\uf8ec\uf8ed1 0 00 1 0
0 0 1
\uf8f6\uf8f7\uf8f8
\u2193 \u2193\uf8eb\uf8ec\uf8ed1 \u22121 20 2 1
0 2 \u22121
\uf8f6\uf8f7\uf8f8 \u2190\u2192
\uf8eb\uf8ec\uf8ed 1 0 00 1 0
\u22121 0 1
\uf8f6\uf8f7\uf8f8
\u2193 \u2193\uf8eb\uf8ec\uf8ed1 \u22121 20 2 1
0 0 \u22122
\uf8f6\uf8f7\uf8f8 \u2190\u2192
\uf8eb\uf8ec\uf8ed 1 0 00 1 0
\u22121 \u22121 1
\uf8f6\uf8f7\uf8f8
\u2193 \u2193
J. Delgado - K. Frensel 80 Instituto de Matema´tica - UFF
Transformac¸a\u2dco Linear - noc¸o\u2dces ba´sicas
\u2193 \u2193\uf8eb\uf8ec\uf8ed1 \u22121 20 1 1/2
0 0 1
\uf8f6\uf8f7\uf8f8 \u2190\u2192
\uf8eb\uf8ec\uf8ed 1 0 00 1/2 0
1/2 1/2 \u22121/2
\uf8f6\uf8f7\uf8f8
\u2193 \u2193\uf8eb\uf8ec\uf8ed1 0 5/20 1 1/2
0 0 1
\uf8f6\uf8f7\uf8f8 \u2190\u2192
\uf8eb\uf8ec\uf8ed 1 1/2 00 1/2 0
1/2 1/2 \u22121/2
\uf8f6\uf8f7\uf8f8
\u2193 \u2193\uf8eb\uf8ec\uf8ed1 0 00 1 0
0 0 1
\uf8f6\uf8f7\uf8f8 \u2190\u2192
\uf8eb\uf8ec\uf8ed\u22121/4 \u22123/4 5/4\u22121/4 1/4 1/4
1/2 1/2 \u22121/2
\uf8f6\uf8f7\uf8f8
Isto e´,
P\u22121 =
\uf8eb\uf8ec\uf8ed\u22121/4 \u22123/4 5/4\u22121/4 1/4 1/4
1/2 1/2 \u22121/2
\uf8f6\uf8f7\uf8f8 = \u22121
4
\uf8eb\uf8ec\uf8ed 1 3 \u221251 \u22121 \u22121
\u22122 \u22122 2
\uf8f6\uf8f7\uf8f8 .
Logo,
[T ]B \u2032 = \u2212
1
4
\uf8eb\uf8ec\uf8ed 1 3 \u221251 \u22121 \u22121
\u22122 \u22122 2
\uf8f6\uf8f7\uf8f8
\uf8eb\uf8ec\uf8ed 3 0 1\u22122 1 0
\u22121 2 4
\uf8f6\uf8f7\uf8f8
\uf8eb\uf8ec\uf8ed1 \u22121 20 2 1
1 1 1
\uf8f6\uf8f7\uf8f8
= \u2212
1
4
\uf8eb\uf8ec\uf8ed 1 3 \u221251 \u22121 \u22121
\u22122 \u22122 2
\uf8f6\uf8f7\uf8f8
\uf8eb\uf8ec\uf8ed 4 \u22122 7\u22122 4 \u22123
3 9 4
\uf8f6\uf8f7\uf8f8
= \u2212
1
4
\uf8eb\uf8ec\uf8ed\u221217 \u221235 \u2212223 \u221215 6
2 14 0
\uf8f6\uf8f7\uf8f8 .
Ou seja,
T(w1) =
17
4
w1 \u2212
3
4
w2 \u2212
2
4
w3
T(w2) =
35
4
w1 +
15
4
w2 \u2212
14
4
w3
T(w3) =
22
4
w1 \u2212
6
4
w2 .
J. Delgado - K. Frensel 81 Instituto de Matema´tica - UFF
Transformac¸a\u2dco Linear - noc¸o\u2dces ba´sicas
c. Verificar que o operador T e´ invert\u131´vel e determinar T\u22121.
Soluc¸a\u2dco: Como sabemos que T e´ invert\u131´vel se, e somente se, [T ]B e´ uma
matriz invert\u131´vel, vamos verificar que [T ]B e´ invert\u131´vel e determinar sua
inversa:
[T ]B =
\uf8eb\uf8ec\uf8ed 3 0 1\u22122 1 0
\u22121 2 4
\uf8f6\uf8f7\uf8f8 \u2190\u2192
\uf8eb\uf8ec\uf8ed1 0 00 1 0
0 0 1
\uf8f6\uf8f7\uf8f8
\u2193 \u2193\uf8eb\uf8ec\uf8ed 1 0 1/3\u22122 1 0
\u22121 2 4
\uf8f6\uf8f7\uf8f8 \u2190\u2192
\uf8eb\uf8ec\uf8ed1/3 0 00 1 0
0 0 1
\uf8f6\uf8f7\uf8f8
\u2193 \u2193\uf8eb\uf8ec\uf8ed1 0 1/30 1 2/3
0 2 13/3
\uf8f6\uf8f7\uf8f8 \u2190\u2192
\uf8eb\uf8ec\uf8ed1/3 0 02/3 1 0
1/3 0 1
\uf8f6\uf8f7\uf8f8
\u2193 \u2193\uf8eb\uf8ec\uf8ed1 0 1/30 1 2/3
0 0 3
\uf8f6\uf8f7\uf8f8 \u2190\u2192
\uf8eb\uf8ec\uf8ed1/3 0 02/3 1 0
\u22121 \u22122 1
\uf8f6\uf8f7\uf8f8
\u2193 \u2193\uf8eb\uf8ec\uf8ed1 0 1/30 1 2/3
0 0 1
\uf8f6\uf8f7\uf8f8 \u2190\u2192
\uf8eb\uf8ec\uf8ed 3/9 0 06/9 9/9 0
\u22121/3 \u22122/3 1/3
\uf8f6\uf8f7\uf8f8
\u2193 \u2193\uf8eb\uf8ec\uf8ed1 0 00 1 0
0 0 1
\uf8f6\uf8f7\uf8f8 \u2190\u2192
\uf8eb\uf8ec\uf8ed 4/9 2/9 \u22121/98/9 13/9 \u22122/9
\u22121/3 \u22122/3 1/3
\uf8f6\uf8f7\uf8f8
\u2193
1
9
\uf8eb\uf8ec\uf8ed 4 2 \u221218 13 \u22122
\u22123 \u22126 3
\uf8f6\uf8f7\uf8f8 = ([T ]B)\u22121 .
Logo, [T\u22121]B = ([T ]B)\u22121 e T\u22121 : R3 \u2212\u2192 R3 e´ dada por:
T\u22121(x1, x2, x3) =
1
9
(4x1 + 2x2 \u2212 x3, 8x1 + 13x2 \u2212 2x3,\u22123x1 \u2212 6x2 + 3x3) .
\ufffd
J. Delgado - K. Frensel 82 Instituto de Matema´tica - UFF
Funcionais Lineares
2. Funcionais Lineares
Definic¸a\u2dco 2.1
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