Synidaemi1 - t s s r q b f h hq p i h a X g b fYY T e d c b...

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Unformatted text preview: t s s r q b f h hq p i h a X g b fYY T e d c b a `Y X WV U T S R @ F E 5 D C B 5 2 7 6 5 4 3 21 6 2 3 1 P 9 I H Q ( ' % ( ' & % @ F E 5 D C B 5 2 7 6 5 4 3 2 1 6 2 3 1 P 9 I H G( ' % ( ' & % @ F E 5 D C B 5 6 A@ 9 8 2 7 6 5 4 3 21 0 ) ( ' % ( ' & %  $ # " !        £ ¦ ¥ ¤£ ¢ ¡¨ £ £   ¢ ¢   © £ ¦ ¥ ¤£ ¢ ¡¨ § £ ¦ ¥ ¤£ ¢ ¡  ¨$$ C ` 4 Y   3$  0  C W G ' & & %  ¨ !  ¨ © D ¨  4 5 G ) 9  X) # ¨ W 4   0  ¨ H $ ¨   ¨    0 © © V   ¨$  (    0 3 0 ¨ ! E G  3  ¨      ©    $$ 2 4  ¨ ! " © ) ( ) ($   # 0 0 © $   ©  9   (     0 E     ¨ H ) ( $    #  0 0  U ! $  ©     ¨ T Q $  © SQ   ©  ¨ $    #   I #  B 0  ¨   ©  $   ©  B 0 G  ¨ !  ¨ © D ¨  4 5   3 9   © ¨ 0 E   © © ¨   ¨ R Q $  © Q   ©  ¨ $    #   0  ¨   B 3 G  ©  $   ©   #P 4 I "  D E   ()  © ¨    ¨    B 3 9 D D    ()  ¨    0  ¨ )  ¨  ¨ H  $$ C   3 0 #   3 0   B ©   ¨    0 0 $ © B  ) ) ¨ $$   ¨ " H 0 ¨ G   0   ¨ )   E 0 9   #   ()  ¨     ¨     0 ) ¨   © $ ¨    ©  ¨ 7     B 3   ¨ 3 0 F 1 4   0   ¨ )   E 0 9   B 3  $ )  $   C    ¨   B 3  "  D $   C   ¨    C    ¨ 0  B A 4   © ¨$   # 0  ¨    # 0 0 ¨ @  0 © ¨  9 8  7 ) 6 5 %   ¨ $$ 2 4    3 2  ( 0    1  © ¨ 0 ) " ! $  0 0 ¨ )  ( ' & & %  ©      ©  ¨$   #  © " !   ¨         ¨  © ¨   © © ¨ § ¦ ¥ ¤ £ ¢ ¡ Q 94   X   #  E        ) ) ¨    H   0 ¨     ¨ H 9 0¨ G  $ C©0 0 ©  # © "©  )  0 ¨ $ )   3 0  !    B ) 3 0  ¨  §6 z= −i ± √ √ √ (−i)2 − 4 · 1 · 5 −i ± −1 − 20 −1 ± −21 −i ± 21i = = = . 2 2 2 2 0 0  $ © B   ) $ ©  ) ) ( A 4  $ C ©0 0 ©   ¨  0 0  0  C X 9  0  $   ©     ¨ H $ ¨ 0   X     3  $ U   (     ¨ § ax2 + bx + c = 0 √ b ± b2 − 4ac x= . 2a   X) ) ¨ H E H ©  # ¨ $  ¨   $   ©   $ C © 0     ¨  4 z 2 − iz + 5 = 0  0 $ U   (    0 X  ¨ § ( ' &%$ #  0  0  C X   # ¨ 0 ! @      √ √ (1 + i)8 = ( 2e(π/4)i )8 = ( 2)8 (e(π/4)i )8 = 24 e8·(π/4)i = 16e2π = 16. ¨ 9 § 4  ¨ E § 4  ¨  0 (  0 ¨ ©  (  ¨   0 0 $ C © 3  0 ¨ 0 ( ' & % $ # 94 8 P7  U Q Q 4 Q 4 T   B A6 4 0   ( E  0 $ C ©  © W ! 5        # 9 H   ¨   0 0   0 4 0 3$    ) ¨ ©  ¨    H   ¨ 0 0  (  "0¨©  (  0  3  0 ¨ $  ¨    $ C © 0 0  © © $ $    ¨  ¨    0    $ ' 0  0   X    ¨  44 Q   0  " 0 © ¨    $ #P ©  B )   1 4   0  (   0  ¨ ©   ( %  0  3  0 ¨ $    ¨  0  $  © 0 0  3 ( ' & % $ # 4  0 0  0  C X E  0    $   $$   0 0 P ! 2      4 ¨ E H   E   9      X$ (   0 I¨)  © 0    0   " X 4 Q 9   © ) 0  D ¨ H    $          ¨ § 4 0  (   0¨ ©  0 ) E © U0   © E 0 (1 + i)8 √ 1+i 2 π/4 1+i = x + yi zp = (21/5 )e(π/5+p·(2π/5))i , π/5 + p · (2π/5) √ 2e(π/4)i p = 0, 1 , 2 , 3 , 4 . 0, 1, 2, 3, 4 p 21/5 −2 z 5 = −2 θ ≈ 56, 3◦ π   1 4 E H 0 0 $ D 9 © ) 0  D  ¨  θ z = 2 + 3i θ = tan−1 (3/2) ≈ 0, 98 z z 1 z w z z+w zw Re z Im z z = (2 + 1) + (3 + (−5))i = 3 − 2i = (2 · 1 − 3 · (−5)) + (2 · (−5) + 1 · 3)i = 17 − 7i =2 =3 = 2 − 3i √ √ √ = 22 + 32 = 4 + 9 = 13 2 − 3i 2 3 z = 2= = −i |z | 13 13 13 2 1 3 = w · = (1 − 5i) − i = −1 − i. z 13 13 4    $ !    ©  ¨$   # 4 Q     0  $  ¨    © ( ) ( ' & % $ # 4 0(  "0¨©  (  0 $ C © G    ¨  3 0    0  3$ C      G 3  0 ¨$ G $ C © ) (    G  © $   ¨ H G  © $  0    G  0 ) ¨ W 4 (   X©¨1 !       z z z = 2 + 3i z z z + w, zw z w/z w = 1 − 5i z  ¢ ¨ ¤  §  £   © ¢ ¨ §¦ ¥ ¤ ¢£ ¢ ¡ %  ¨  9 H © 0 0 I  0 0  09$  0  (  $$  E  0  ( ¥ 4 $ © E      $  0   0 0 ¨ ©  $    09$  0  (    0 ¨  ( $ © E      $   0  09$  0  (    0 C$   0 1 4 9 3 0 #  D D   0 )¨ ©6   0  © ) 0  D 9  V   0   0 0 ¨ ©   © ) 0  D 0  (     ( Q V  0 3  0 ¨$   $    9 H   ) ) (   "    ¥ 4   B 3    ¨ H 9  $ E   ) ) ¨   © D  )   0    0  0 ¨ © 3  0 ¨ $     $    ©  " X0 ( ' & % $ # 9 8 2 17  U R4 %4 Q   B A6 4  $  E  09$  0  (   (   0 0 ¨ © 9   0 0  09$  0  (    0 C$ $$  E  0  (   0 0 P ! 5 2     4 $© ¨ 0 ¨     ¨       © 2 (0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 1, 0), (1, 0, 1), (0, 1, 1), (1, 1, 1) (0, 0, 0) (1, 1, 1) (0, 0, 0) (1, 1, 0) (1, 2, 3, 4) (1, 2, 3, 4) (1, 2, 3, 4) · (1, 1, 1, 1) (1, 2, 3, 4) (1, 1, 1, 1) √ = arccos ( θ = arccos  ¨  0 0    $$  E  0  ( ¥ ( ¨ U ) (1, 2, 3, 4) = 30 √ 10 30 · 2 = arccos(0, 913) = 24, 1◦. (1, 1, 1, 1) = √ 4 = 2. (1, 2, 3, 4) · (1, 1, 1, 1) = 10 v·w vw θ = arccos .  0 0 $ U   (   ¨   I ¨   ¨  (     X  ¨ © $$  E  0  ( ¥ ( ' & % $ # 49 (  0 0    $$   0  (   0 0 P ! 2 2     4  0 ¨) ¨   ¨ 7   D D       D 0   $ 0 3 0  © (    " D   E     X0 ¨   ©  ¨  4 0    $   ¨ $  3 0 ¨ V    3 0      ¨    H    ) ) ¨ 0 ¨   3 0      ¨    H  ¨  0 U   © F$  E H 0 $ ¨ $ © ¨    0 F  0   0 0 )¨ 4 4 4 ¨4 H G )  ! $©   0  0  $  0 0 I   $ ©  (   © ( 0 ¨    0 0 ) ¨   © " X$  ) )¨   3$¨   ¨    B 3    C  A 4   0 3 0     (   0     B ©   © ) ) ¨ H "    ¨ H   © ( 0  ) ) ¨    ¨  9   0  C X   ¨6  0  C X E H       D 0    $  0 3 0   0  © (    ¨$ )  ©   V      ©      ©$$    H   ) $ ©   )9$  A 4  (   0    B ©      ) E   $ ©  ¨  H   0  0  C X    B ©    3   B 3    ¨ H A 4 ©  ¨$   C   ¨  ¨  E  3$   0 ¨  ¨  (  0  ¨ ©    C  9  (  9      0       ¨ E H  ¨   U 0   © ) #$  4 ¨ E H  1 4  0  ¨ ©    C  9  (  9       ¨$ )   0        E H  ¨         )   U   0      © ¨     ¨  © "  E   ©  G           ¨ © ¨  0  ¨ ©   C  9  (  9       ¨     ¨  0     § ¦     0 ¨  U 0    C ¥ 4 (  ©$ $    ¨ G  © B E § 4  0 0 C X 9 0 0   X©¨ 1 θ v w (1, 2, 3, 4) (1, 1, 1, 1) R4 c c = −2 (c2 , c3 , c4 ) = (4, −8, 16) = 4(1, −2, 4) −2 c c = −2 c = −2 c3 = −2r c2 = r r c = −2 cr = −2r c3 = −2r c2 = r c = −2 c4 = 4r. ¤   0  C X  E X H   E P 4   0 0  H  $  © $ ©  ¨ E H  0  ¨ ©    C  9  (  9       ¨  ¨ ©  0      1 ( ' & % $ # 9 8 P7  U S % 4 Q 4 Q   B A 6 4  0  ¨ ©    C  9  (  0   9     0 0       0 0  H  $ C ©  $$   0 0 P !  2     (c2 , c3 , c4 ) = r (1, −2, 4) r>0 (1, −2, 4) (c 2 , c 3 , c 4 ) c  ¢ £ ¢ ¢£ ¡ ¢ ¨ §¦ ¥ ¤ ¢£ ¢ ¡ R    #  E     ¨   ( 9      ¨  ©E 0 4 &  ) )¨ ¨    3$     0 0  H     $$      E      0 F        H   4    ¨  0 I B    ©     ¨ §   ( ' & % $ # 4 & 0  C X  0     3  0 ¨$  ¨ E H 9       0  $ 0   X   ¨  (  1 4 0 C X  ©$$  ¨  0  E 0 3  0 ¨ 0  4   ¨ ©  (    0 0  3  0 ¨ $ 0  C X  ©  ( )  ©  0 0      ¨ E 0  3  0 ¨ 0   4 & 3  0 ¨$  ) ) ¨    ¨  0 0    $$ U 0  ) ) ¨  ¨  ¨  9      4    0    ¨   ©©V  ¨    0  #$$   3 0  ©¨ © (    $ ©   ¨ 1 !  2    4   0 © ) 0  D 9 0  (  © © V     ¨     0 0  #9  H  (  © © V  0  (    ¨  (  0      v v = (v1 , v2 , . . . , vn ) R n v=0 v w v v v−w Rn cv cv v+w v w Rn − → AB − − → BC B → −− →− AB · BC = (1, 3, −2) · (4, −2, −1) = 4 − 6 + 2 = 0.     E X1 4  ) )¨   0 0  ©©V 0 (  ¨ H  ¨ E H &  ¨  3$ ¨   $ D ¨ 3 ¨ @    ¨ H  3$ ¨     $ D ¨ 3  0 ) ¨    9 H  ¨   © © V  0  (   V       ¨ © © (        )   U    0 C    4 ©©V 0 (   ¨       ¨ H ¨ ©  X ¨  0¨ ¨  0 ¨   ( ¨   3 0 #©©V  ¨ 0 0    0 0  # 9  § 4  0   0 0  #9  H $   0 0 ¨  9          ¨    ¨        ¨ § − → AB = (4, 3, 0) − (3, 0, 2) = (1, 3, −2) − → AC = (8, 1, −1) − (3, 0, 2) = (5, 1, −3) − − → BC = (8, 1, −1) − (4, 3, 0) = (4, −2, −1). C = (8, 1, −1) ¨ ( A = (3, 0, 2), B = (4, 3, 0)  0       () 1 ( ' & % $ # 4   0 0  # 9  H  3 0  #  © © V    © ) 0  D 0  (   0   © ) 0  D    0 F 1 !  2     θ = arccos(− 1 ) ≈ 109, 47◦. 3 (1 x·y cos θ = = xy 3 4 1 −4 3 4 = 3 4 −1 4 = −1. 3   0 0    $$  E  0  (     3$  E H Q Q4 %   0 0 ¨  $ ) 1 ©  B )   1 4 ¨ 9  0   " ©   0 © ¨ $ ©  0  " ©   0  ¨$ ( ) E   0 0     4 ¨ 9  0  " ©   0 © ¨ $ © 9  0  " ©   0  ¨$ ( ) E   0 0 "    4     © ¨$   ¨   0  (    ¨ 0  0 0    (   3 0 #         ¨ H  ¨   " 0  ( ¥ 4   0   " ©   0 © ¨    X  ¨ © $ ©  0  " ©   0  ¨$ () E        0 0 P ( ' & % $ # 9 8 P7  U % %4 %4 Q   B A6 4  0  © ¨  9   0  (   X 0 ¨ ©  0 ) ¨ W 4  3 0 #  0  " ©  0 ) E ©  ¨  0 0  © ) 0  D $$  E   0 09$  ¨  0 (  4 4 4 ¨4 H G   0 0 ©¨   ¥ " " ¥ 9  0 (   ¨ ©¨ '§¦ ¥ $ ¤£'¢¡ 4  0 © ) 0  D 9   " ©   0¨$ () (  0  © ) 0  D 9 V 0  " © 0©¨    0 0  H 4 34 © G  0  0 © C 7 " X  X  9 V   "©  0¨$ ()  (   0  © C 7 " X   ¨$ $ ¨   © ) 0  D 0 (  9  V    C X 0  "©  0 ©¨    0 0  H ¥   " © 0 ©¨   3 0 #   V E   ¨ Y ! @ 2     1 (2, 1, 1) = (1, −1, −1) 22 2 2 2 1 y = (− 2 , 1 , − 1 ) 2 2 θ 11 (2, 2, 1) 2 x = (1, 0, 0) − (0, 1, 0) (1, 0, 0) 1 (2, 1, 1) 22 (1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 1, 1) 4 2 = arccos √ √ 32 ( (1, 1, 1) (1, 1, 0) √ 2 √ 3 = arccos = 35, 3◦ . Q Q4 %   0 0 ¨  $ ) 1 ©  B )   1 4 θ = arccos (1, 1, 1) · (1, 1, 0) (1, 1, 1) (1, 1, 0) ¨  0 0    $$  E  0  (   0 ) ¨    θ S 4 & 3  0 ¨$  ) ) ¨  ¨  0 0    0 ¨ Q 3  0 ¨$      (  0       E  4 ( $     © ¨   V ¥ 4   ¨  0 I B    ©     ¨ §   4 0¨ ¨ E § 4 (   X©¨ 1 4   ¨  0 I B    ©     ¨ §   4 & 3  0 ¨$    ¨   ¨  0 0    $$ U 0  ¨ $  G   B 3 0 ¨ 3  0 ¨$   E !      (  0      4 ( $ © G    © ¨   V ¥ 4  ) ) ¨ 0    ©    0 0 ¨  3$    0   3 0 ¨   (   ¨    H   B 3   ¨ ) E  0 0 I  © ¨ !  ¨ E H  0 C V  0  #$$   ¨  0 ¨  E    0 F   X$    ¨ § 4   ¨  0 I B    ©     ¨ §   4     © ) #$ E U 0   © ¨ Y 4  ¨ 9 H  ( & $$ C ))¨ 0 © 0    ¨ E H 0 0    $$ U 0  ) ) ¨  ¨  ¨    § 4  ¨  3 0  ¨   $  )  E 0  3  0 ¨ $  ¨ '4 %   0  0  ¨   $  ) 1 ©  B )   1 4 0 0      $ $ U 0  ) ) ¨ V  v = (1, 0) v − w = (1, −1) ¡ v = (1, 1) w = (0, 1) √ c v =− 2 ¡ w = (−1, −1) + 2 v2 v v w cv = [−1, −1] = w ¡ c = −1 v+w v = (1, 1) 2 v1 +···+ v 2 2 2 v1 + v2 + · · · + vn > 0 √ 2 √ 2 v v1 , v2 , . . . , vn v >0 2 vn ' x1 = 1−2s +t x2 = 2 −t x3 = 1+2s+3t. ¨  0  $ D   #    0  C X  © 0 ¥ !      §    §                      © 4    ©  ¨$ "  # 4 & %  0  ¨   $   0 ¨ © 9 3 0 F   ¨   ¨    U 1 4  0  $ D E     ¨ H    0 0 I E H 0 ¨ ¨$   C   0 0¨ ¨ 0 ¨    X § 4  ©¨$ # 4 S  0¨  0 $$ C X    ¨   H 0¨  D D ¨ 0  0 C X   # ¨$      H  0 0   ¨     0 0 ¨  A 4  0 0  © $   0 0    # ¨$   $ ©   ¨$  B ©    () 1 x = x0 + su + tv = (1, 2, 1) + s(−2, 0, 2) + t(1, −1, 3). 9 H  ¨ 0  0   X) © 1 4 9 )9$   H 0 0 © ) 0  D ) ($  ¨ E H 0 0 $ D 9 ¨   © ) 0  D  D D  ¨ 4 4 4 ¨4 H6 0 0 $ D 9  X $ u = (−1, 2, 3) − (1, 2, 1) = (−2, 0, 2) v = (2, 1, 4) − (1, 2, 1) = (1, −1, 3)  0     4   X©¨ 1 4  0  0  $ D 9  X  $  ¨        ¨  (  (  0  0  $ D 9  ¨  ¨   © ) 0  D © 0    ¨  ¨    H x0 x0 = (1, 2, 1) u v x = x0 + su + tv, 0  ( E ¨  0 $ D  0 X) ©1 4 0 C X) ©  0 0 I   9 H E   X #  ( ' & %$ # 4  0 0 $ D 9 0C X )9¨ © )6 0C X  0 0 ¨  0 0 P 4 (  0  © ) 0  D      $  ¨   0  $ D   #  0  C X) ©  0 0 P ! 2 5     (2, 1, 4) G (−1, 2, 3) (1, 2, 1) x1 = 1 − s x2 = s + t x3 = −s + t.    E E § 4 © 0  © ¨   # 0C X V  ¨   0 $ D     X©¨ E   0 0 1 x = x0 + su + tv = (1, 0, 0) + s(−1, 1, −1) + t(0, 1, 1).  0  0 $ D  0   X) ©   ¨ E § 4 4 34 © G 0 0 $ D 9    $  ¨ © ) 0  D 0¨  0 ¨  ¨   ©E$ ( 1 4  9    ) )¨  ¨$ " X$    ¨      ¨ ©  ¨ § x0 = (1, 0, 0) x0 u = (0, 1, −1) − (1, 0, 0) = (−1, 1, −1) v = (1, 1, 1) − (1, 0, 0) = (0, 1, 1). 4 34 © G 0 0$ D 9  © ) 0  D  3 0¨   E !  ¨      0 0 I   9 H  ¨  © ¨     © ¨    § 4 ) " ! 9 4Q 4 $ ! G 4 ¨$ D   ¨ 1 9 3 0 F  ¨   ¨      ¨ § 4   9      ) ) ¨   ¨ 0 ¨  0 0  $ D 9  X  $  ¨      ( ©  0 0 I        § ( ' & % $ # 4 ( G  0  © ) 0  D   0  ¨  9    $  ¨  9  0  $ D   #  0  C X) ©   0 0 P !  5     (1, 0, 0) (0, 1, −1) (1, 1, 1) R3   ¤ ¤ § ¦ ¢ ¥   ¤ ¤ £ ¢ ¨ ¢ ¡ ¢ ¨ §¦ ¥ ¤ ¢£ ¢ ¡ $ x 4 − 2x − 3y 4 −2 −3 y = = 0 + x 1 + y 0 . x x y 0 0 1   E   (    (     E D D   © © ¨ H   © W 4  ¨ E § 4 ) " ! 9 R4 Q 0  ¨     0  ¨ 9  ¨ 3 0 F   ¨    ¨        ¨ H  ¨   0 0 0 C X E  0 0  $    0  )    ¨  0 0C X    0 0 I  $ ©  ¨$ 3$ C 0 1  4 0 0 $ D 9   ¨$ ) ¨   © ) 0  D  ¨ H © (  "D   ©  $ E X  ¨   0 3 0 ¨  2 4 0 0 © ) 0  D   ) ) (   ¨   ¨  ¨  0 0 0C X 0  0 x = 4 − 2y − 3z x+2y +3z = 4 1 t = −3 2 (3, 2, 2) 33 4 = x + 2y + 3z = (1 + t) + 2(1 + t) + 3(1 + t) = 6 + 6t.  E (  0 0C X 9 0 0  X©¨   9 H ¨  0 0 $ D 9  ¨   (  ¨ H E   © ) 0  D ¨  0¨ © (     ©   4    (       0 0  09$ E   © ) 0  4  0 0C X $$ # D D   ¨    © ) 0  D   ¨ H   ¨ $  B ) E 0  ¨  0  0  $ D 9   0   © ) 0    ( ' & % $ # 4  0   0  $ D  0  C X  )  ©    0 0  P    4   0 0  09$  (  0  0 $ D © ) 0  D    )   0 0 P  4  0 0 C X  ¨  0 $ 94 R & & %  ( I " D  6 ! 5 5    # x + 2y + 3z = 4 (x, y, z ) (x, y, z ) = (1 + t, 1 + t, 1 + t) # x + 2y + 3z = 4 (x, y, z ) = (1, 1, 1) + t(1, 1, 1) # " x1 + 4x2 + x3 = 10.   E ( ( d (a, b, c, d) = r (1, 4, 1, 10) ¨ ` 4 0  0 X © B 0 0  X©¨   9 H   #  D D ¨ 0   0  C X  © © ¨ H    # ¨ 0 r=1 a, b, c a+2b +c = d −a+2b+3c = d 2a +b+4c = d.  D D ¨ 0   0  C X E H   E P 4  (   # 0 0   ©  9 0 0 0 0 $ D E  X $ ¨  X  H  0 0  © ) 0  D © 0    X©¨    ¨ H ©©V  ¨   E 0  0 4  0   (  E  0  C X     © ¨ 0 !    ¦  §         x1 , x2 x3  ax1 + bx2 + cx3 = d !  x1 + 4x2 + x3 = 10. 9 H ¨ 0 0 4 0     (    X  ¨$     ¨ § 4  (    © B   $   ©     ¨ H  ¨    0 0 C X  ¨  D D ¨ 0  0 C X   ¨ H E 0   $ 0 1 a = 1, b = 4 d = 10 c=1 a − b + 3c = 0. (  § ©  ©  £ ©   ©  £  ¥£  £ © § £ ¨ § ¢ ¦ ¥ ¤ £ ¢ ¡ −2a + 2c = 0    3$      H 9 § ax1 bx2 cx3 ax1 + bx2 + cx3 = a(1−2s +t) = b(2 −t) = c(1 +2s+3t) = d +0s+0t E ©     ¨   X$  4  D D ¨ 0  0 C X D D   X©¨   ©  B   ¨  (  0  ©   0 0 I   $ 3 4 0   ( E  0  X   3 0 ¨ ©  © ¨  ( 0      ¨ ©  $    ¨ H  0  ) ©     ¨  E H  3$  © © V    ¨      0   ©    1 4 0      3$ ¨         ¨ H 0  9  X ¨$  (  ©    ¨$ I B  ¨  0 C X  X ¨   3$      9 H E    #! 0 ¨  2 a, b c ax1 + bx2 + cx3 = d 4 (    0 F     ¨ 0 0   ) ) #0  ) ( 0 4  0 0      © © E  ¨    H 9 H   ©  B 0  ¨  ¨      0 ¨  U 0    C ¥ proju y = proju x projv y = projv x. x = proja x + proju y + projv y, y = proju y + projv y.    3$  $ "  ©  B )   1 4 9     $     © ) #$ E   ( E   ©©V 0 (   ¨ %4 R   0 0 ©¨ 1 ©  B )  1 4   X©¨1   y = x − proja x y P y a x = su + tv = proju x + projv x. ¨ 9 H ( © ¢¡ projv x = tv  © B ©©E     x·u u u2 (su + tv) · u = u u2 su · u + tv · u = u u2 su · u u = u2 su2 = u u2 = su. u·v =0 proju x =     E  ( 9 ) " ! 9 S% 4 $ ! E X6 %4 R   0 0 ¨ $ ) 1 9 $ U  (   ) ) (   © ( ) 4 E D   0   (  0 ) ¨   ( U 0   © W 4 0   ( E 9 © ) 0  D 0  ¨   V  ©  E   (     0 0  D   ¨  (    ¨    0  $ D  ¨    §   ( ' & % $ # 4  © © V 0  (   ) ) ¨   ¨  (  ¨  3 $      ) ) ¨    H   $ "   9 0    ©      0     B 3  ¨    0 F 1    u P P su + tv (0, 0, 0) x = su + tv u u x v v x = proja x + proju x + projv x.    3$  9    0  ¨   V       0 F 1 4      #   E     ¨ Y      3$  9    $  ¨  © ) 0  D 0  ¨   V       0 F 1 4      #   E     ¨ Y   94 8 2 17  U Q Q4 R4 Q   B A6 4   0  $ D E      ¨ H V     (  (   0        0 0 D ¨ ¨  0  ¨  9 0 $ D V    #  E     ¨ Y ! @ 5     4 Q4 R   B A 9  ¨ © ¨   ( ©©E      E  0 0 I (  E   0 0 C X    4 ( G  0©)0D  ) ¨ ©   B 3 $ ©    © B    B 3    ¨ H A 4  0 0  09$   C  E  ) ) ¨  V  9  H $$     0 0  H 0 0 $ D 9  X $  ¨  © ) 0  D E X H  0 0 I    ¨   B 3  ©©¨ H  © E   $© ¨$  0 0 u·v =0 x R3 x = proju x + projv x. u v a u·v = 0 P x P ⊂ R3 (0, 0, 0) 4 (0, 0, 3 ) (4, 0, 0) (0, 2, 0) 5 4 $ "  9  0   C ©    0  0 0     $ ©  ¨$ 0 #    0 9 H ¨ ©©V 0 (   V  (     0  3 0¨(P 4 ¨ U ) (     E   (  D   0   (   #  0 $ U  (   © ( ) 4  © © V 0  (   ) ) ¨  ¨  (  0      ¨ 0 $ "   X©¨ 1 4  0 ¨    9  ¨ ©  F $  (     U  U 0  ¨ 0    © 1 4 9    $  ( (     0 0  D   ¨  ¨  9   0  $ D    ( ) 1   u v u = (1, 0, 0) v = (1, 1, 0) projv y = projv x ¨ E § 4 x = proju x +projv x 1 projv x = ( 2 , 1 , 0). 2 proju x = (0, 0, 0) (0, 0, 0) u v R3 x = (0, 1, 0) x = proja x + proju y + projv y = proja x + proju x + projv x.   © B  © ©E      E  ( proju y = proju x   ©0 F U0   C ¥ y·u u u2 (x − proja x) · u = u u2 x · u − proja x · u = u u2 x·u u = u2 = proju x. proja x · u = 0  9 H proju y =   © B   #  0 $ U  ( D D    X©¨     ¨ § 4 E  © © V  0  (   (  0       E !   ¨ 9 H  (    3$ ¨       ¨   D D    X W proja x proju y proja x a u v T 4  D D ¨ 0  0C X  ¨ 7  D D  9 0 0  X©¨   9 H ¨   0 0  $  " D    ¨ ©  $E X1 x1 −1 + 2x3 −1 2 x2 = −1 + x3 = −1 + x3 1 . x3 x3 0 1 0   $   0 ¨ E H   C ¥ 4  ¨  0 0  09$    ¨          ¨  0  0   4  ¨  0    (   $$  ©  3 3    09$    ¨ 0          ¨  0  0   4 $ © ¨¢§ § (  G  0 0  © # ¨  !          ¨  G R  )$ E 3 9 $$    © 3 0 ¨ 0 0  0 ¨  ¨   § 1 0 −2 −1 0 1 −1 −1 1 −3 1 2 0 1 −1 −1 ¨ 9 H (  ¨ G ; x3 x3 x2 − x3 = −1 x1 − 2x3 = −1 &¦ ¥ ¤§ ¤  ¨  x2 = −1 + x3 x1 = −1 + 2x3  ; 1 −3 1 2 3 −8 2 5 L1 + 3L2 L2 − 3L1   (   $$  ©    ¨ © ©   E    $ ©    ¨     09$   © ( 0  (  )$ #  0 )   D D   $$ © 1 ( ' & % $ # x1 −3x2 +x3 = 2 3x1 −8x2 +2x3 = 5.  0  D D ¨ 0   0  C X  0    $   $$   0 0 P ! 2 @     x = (x1 , x2 , x3 , x4 , x5 ) = (5 + 2x2 − 3x3 − x4 − x5 , x2 , x3 , x4 , x5 ) = (5, 0, 0, 0, 0) + x2 (2, 1, 0, 0, 0) + x3 (−3, 0, 1, 0, 0) + x4 (−1, 0, 0, 1, 0) + x5 (−1, 0, 0, 0, 1). x1 = 5 + 2x2 − 3x3 − x4 − x5 ¨ 9 H (   ¨  x1 − 2x2 + 3x3 + x4 + x5 = 5 0 0  x = (x1 , x2 , x3 , x4 , x5 ) = (2x2 − 3x3 − x4 − x5 , x2 , x3 , x4 , x5 ) = x2 (2, 1, 0, 0, 0) + x3 (−3, 0, 1, 0, 0) + x4 (−1, 0, 0, 1, 0) + x5 (−1, 0, 0, 0, 1). ( ¨ E § 4  0  0  C X ©   #     # ¨ 0 4 0   (    V   (    0  ¨    C P   4  0 0  0  C X E 0    $  0 0 ¨ $   G  0   G 0  $ V  © B !    ¨  E  E   0 0   0 x1 = 2x2 − 3x3 − x4 − x5 x1 − 2x2 +3x3 + x4 + x5 = 0 x1 + 5x2 − 2x3 = 0 (5, 0, 0) x = (x1 , x2 , x3 ) = (5 + 2x2 − 3x3 , x2 , x3 ) = (5, 0, 0) + x2 (2, 1, 0) + x3 (−3, 0, 1).    © B  0  9 1 4    0 ¨   (  0 0  0  C X  U © U     0  0  ¨     © ¨   0  0    $  0 0 ¨  $  E    $  3   x1 = 5 + 2x2 − 3x3 x1 x = (x1 , x2 , x3 ) = (2x2 − 3x3 , x2 , x3 ) = x2 (2, 1, 0) + x3 (−3, 0, 1). ¨ E § 4     E   (  0 0  0  C X 9     0  0  1   ( ' & % $ # £   0 0 0 C X 0   $ 0 0 ¨ $   ¨  ¨ ¥ 4   0 0  0  C X 0    $  0 0 ¨  $    0 0  P   4  0 0   $ 0 0 ¨ $  (   0 F  (  0 0  0  C X E 0    $  0  ¨   0 0  P   4 9 ) " ! 9  ( © ) 9$     X  ¨ © © ) ¨ © 0       ¨$ 09$  ¨6   0 0  0  C X 0    $  0 0 ¨  $    0 0  P   !  @     x1 = 2x2 − 3x3 x1 x1 − 2x2 +3x3 = 5 x1 − 2x2 + 3x3 + x4 + x5 = 0 x1 − 2x2 + 3x3 + x4 + x5 = 5 x1 − 2x2 + 3x3 = 0 £ ¤ ¤ ¥  ¦ ¨  ¢ ¡ ¢ ¥ ¤ ¨ ¢ ¡ ¢ ¨ §¦ ¥ ¤ ¢£ ¢ ¡  0 0  0  C X  X) $ #  E  0  0    $  0 0 ¨  $    0 0  P ! 5 @     1 2 0 −2 x1 2 0 −11 x2 6 −6 2 2 −4 −4 x3 x4 2 −7 −8 −11 18 = 6 7 4   (   $$  ©    ¨ © ©   E  I # 0 )$ #  0 )     ()   $ ©   0  #¨    Y   ©¨ !  (  )$ #  )   D D    X©¨ 1 ( ' & %$ # 1 2 0 −2 1 2 0 −11 −11 0 6 −6 2 18 ; 0 2 −4 −4 6 2 −7 −8 7 0 1 0 ; 0 0 1 0 ; 0 0 1 0 ; 0 0 1 0 ; 0 0 1 0 ; 0 0 1 0 ; 0 0 1 0 ; 0 0 2 0 −11 −11 2 −6 24 40 2 −4 −4 6 6 −7 −30 −15 L2 − 2L1 L4 + 2L1 2 0 −11 −11 2 −6 24 40 0 2 −28 −34 L3 − L2 0 11 −102 −135 L4 − 3L2 2 0 −11 −11 1 1 −3 12 20 − 2 L2 1 0 1 −14 −17 L 23 0 11 −102 −135 2 0 −11 −11 1 −3 12 20 0 1 −14 −17 00 52 52 L4 − 11L3 2 0 −11 −11 2 −6 24 40 0 1 −14 −17 1 00 1 1 L 52 4 2000 L1 + 11L4 1 −3 0 8 L2 − 12L4 0 1 0 −3 L3 + 14L4 0011 200 0 1 0 0 −1 L2 + 3L3 0 1 0 −3 001 1 L1 − 2L2 000 2 1 0 0 −1 0 1 0 −3 001 1 4  0 0 C X   ¨ 7  D D  9 0 0  0 0  $  X©¨  ¨    0 ) ¨ V ©©V  " D (   ©¨     ` 4 9 H  ¨  0 0 0 C X E 0 0   0 (x1 , x2 , x3 , x4 ) = (2, −1, −3, 1) &Q QQ 4   ( !  #¨ 7  ¨ H E $$   X $ 9  H  0   © ) 0  D         )   U   0      ¨ ©$$ ¨   2 4  0  0  C X 9 H    ¨   0  B 3 9 0 0  ( !  # ¨$ P 4  (      $  y = 3x2 + 5x a = 3, b = 5 c=0 1 11 8 L2 4 −2 1 2 ; 4 1 11 8 2 1 22 L1 4 2 1 22 4 −2 1 2 1 1 1 8 L2 − 4L1 ; 0 −2 −3 −10 0 −6 −3 −30 L3 − 4L1 8 1 1 1 ; 0 −2 −3 −10 0 0 6 0 L3 − 3L2 1 1 1 8 0 −2 −3 −10 ; 1 0 0 1 0 L 63 1 10 8 L1 − L3 0 −2 0 −10 L2 + 3L3 ; 0 01 0 1108 1 − 2 L2 ; 0 1 0 5 0010 1003 L1 − L2 0 1 0 5 ; 0010   E P 4   ¨    09$  ¨   ($$  ©  ¨ E  I # 9 H    ()  (  )$ #  )  D D    X©¨ 1 4a−2b+c = 2 a +b +c = 8 4a+2b+c = 22.  D D ¨ 0  0 C X   E P (   0   0  C X © E  ( ©  0  © ) 0  D  0     ()      ¨ § a+b+c=8 4a + 2b + c = 22. 2 = a(−2)2 + b(−2) + c = 4a − 2b + c.  X ¨     ¨   H G © U E        © © B E H  0   ( !  # ¨ 7 0 C X 9 0 0   X©¨  ¨ (   0   ( !  #¨ 7 E ¨ 0 0   © ) 0  ( ' & % $ # 4 (  0© ) 0  D  0 ¨  9  0 0  ( !  # ¨ 7    0 0  H  (  $   ©   0 0 P 9 R & & %   ( I "  D  6 ! @ @     y=2 (−2, 2) x = −2 a, b (−2, 2), (1, 8) # y = ax + bx + c 2 c (2, 22) " %Q 4 0   $ 0 ¨   ¨$  B ) E 0 ( ¨ 9 § x1 = 1 − x2 = 1 − 1 a−2 = a−3 a−2  ¨ § 4 1 a−2 +x2 = 1 (a − 2)x2 = 1. a=2   ©$$  9 H ©  B  x2 = x1  0  D D ¨ 0   0  C X  0    $  (   C   B H   ¨  0  D D ¨ 0   0  C X   ¨ 7   D D   0   $    0 F  0   (    V   0    0 0 ) ¨  © £  ¨ ©  ¨    ¥ 4 ¨ 0   $   0 ¨    ¨   D D ¨ 0   0  C X   9 H   © ) #$  4 ¨ V  D D    ¨)   © ¨ § 4 &  ) ) ¨  ¨  ¨   $  ©  ¨  0 ¨  ¨    B  $ ©   0 )$ E 3 9 0 ¨ $    © 0 9   " &    " ©0 ¨  ¨   09$  ¨   H   9 H E  0   (   $$  ©    ¨     H ©  V  0    $   0 ¨    ¨   D D ¨ 0   0  C X    ¨ § " a=2 a=2 a−2=0 1 2 1 a 1 3 ; 1 1 0 a−2 L2 − 2L1 1 1 4  ($$  ©  ¨ E 9 H    ()  (  )$ #  )   D D    X©¨ 1 ( ' & %$ # 4   0    $         ¨ $ 0  3 0 ¨ "      ( 0    $  0  ¨   ¨ $  B ) E 0    G 0    $   0¨   I   D D ¨ 0  0 C X    0 0  H  0 0 $ C© E 9$ ©  ¨  ¨  0¨ ¨6  3$  $$ C  0 0 P a x1 +x2 = 1 2x1 +ax2 = 3.  D D ¨ 0  0 C X  3 0     © ¨    () 1 ! 2      4 0    $   0 ¨  D D ¨ 0   0  C X   ¨ 7   D D     ¨  9 § 4  0  C X E H  $$ # D D     ¨  ¨$ 0  0 1 0  0   X  3$   0 D D ¨ 0  0    $       ¨  0   (    V   0 )$ # ) ($ 9 0  09$  ©  ¨ ) 0x + 0y + 0z = −1. 1 1 1 1 1 1 ; 0 −2 3 −2 0 2 −3 2 1 1 1 11 ; 0 −2 3 −2 0 0 0 −1 1 1 1 6 3 −2 "0C X  ¨ 7  D D  E (x, y, z ) 1 3 1 L3 + L2 L2 − 3L1 L3 − L1 ¤  ¨ 0  0 0 ¨ H E   ¨   B 3 $ ©  © B   0    0 0 ) ¨  © 4     ¨     09$  ¨  9 H 9 ©     X  (  )$ #  )   D D    X©¨ 1 4  D D ¨ 0  0 C X   #¨$    0 #¨ W ( ' & %$ # 4 0 $   0¨ I  " x +y +z = 1 3x +y +6z = 1 x+3y −2z = 2  D D ¨ 0   0  C X   E      0 F 1 !        £¤ ¤ ¥  ¦ ¨  ¢  ¡ £ ¢ ¢ ¥ ¤ ¨  ¢ ¤ £  ¦ ¨ £ ¤ ¢ ¨ §¦ ¥ ¤ ¢£ ¢ ¡ ¢ £ ¢£ ¥ ¤ £ ¢£ ¡ 4 &Q4 '   0 0 © ¨ 1    0 " © ¨   ( E    0 0 $  X©¨  (  D D¨ 0  0 C X   ¨$ 09$  #¨ 0 ! 5      x1 −2x2 +x3 +x4 = 4 2x1 +x2 −3x3 −x4 = 6 x1 −7x2 −6x3 +2x4 = 6. 94 8 P7  U ¤ 9 H 9 ©    "     Y  (  ) $ #    )   D D    X© ¨ 1 ( ' & % $ # R4 4 Q   B A6 $ $ 4 1 −2 1 14 1 −2 1 1 2 1 −3 −1 6 ; 0 5 −5 −3 −2 L2 − 2L1 1 −7 −6 2 6 0 −5 −7 1 2 L3 − L1 1 −2 1 1 4 ; 0 5 −5 −3 −2 0 0 −12 −2 0 L3 + L2 1 −2 1 1 4 3 1 L ; 0 1 −1 − 5 − 2 5 52 1 1 00 1 0 − 12 L3 6 5 1 −2 0 L1 − L3 4 6 0 1 −0 − 13 − 2 L2 + L3 ; 30 5 1 00 1 0 6 16 1 1 0 0 − 30 5 L1 + 2L2 0 1 −0 − 13 − 2 ; 30 5 1 00 1 0 6 ¨  © E   ) ) ( 0 D D ¨ 0  0 C X E 0  $  X ¨  V1 4 9 D D ¨ 0  0 C X    $  " E 0 $6  ( 9 0  $ V6 U0   X©¨ 1 4 ( G ¨ E § 4 © #¨ ! $ E X  ¨     ¥ x1 = u1 = ( 16 , − 2 , 0, 0) 5 5 x 16 5 2 −5 2 1 13 + 30 x4 x2 = − 5 + 30 x4 1 v = x4 ( 30 , 13 , − 1 , 1) 30 6 1 30 13 30 −1 6 16 5 2 −5 + x4 . 0 0 1     ¨  (   0    $   (  ©   3$   0 ¨ E § 4 x4 = 30r x = u1 + v = 16 5 1 + r 13 . x= 0 −5 0 30 RQ x4 1 x3 = − 6 x4   X©¨ 1 b1 , b2   $ ©  ( SQ 4 0  $   C   I # I    D D ¨ 0  0 C X    3 $         H    #$  )     ¥ £ 0    $  0 ¨   ¨$  B ) E 0 I   b1 , b2 , b3 b3  D D ¨ 0   0  C X    0 0  H x1 +x2 +x3 = b1 2x1 −2x2 +4x3 = b2 5x1 −3x2 +9x3 = b3 .   0 $ C©  X$ ¨   © ¨$   C   1 2      4 3 2 3 4 5 − 5 x3 −3 3 3 1 = 2 + x3 − 1 . − 3 x3 3 3 x3 0 1  ¨ 0 U ( − 5 x3 3 x3 = 2 3 − 1 x3 3    ¨ H 0 0    0 4 4 3 a=8 ¨ x1 x2 = x3 x1 = 9H   ¨  0  0  X   1 x1 5 4 + x3 = 3 3 1 2 x2 + x3 = . 3 3 ¨  D D¨ 0  0C X  3 0   1 1122 2 1 1 2 0 −3 −1 −2 ; 0 1 1 2 3 3 0 0 0 0 0000 5 1034 3 1 ; 0 1 3 2 3 0000 a=8 L1 − L2 . 1 − 3 L2 © B  ¨ § 4 &Q4 4  ! G 0  $  0 ¨   !  ¨   D D ¨ 0  0 C X  (  )$ E 3   X ¨  V 9  0 0 D  ¨ E H    ¨ § $ a=8 1 122 2 1 1 2 2 −1 3 2 ∼ 0 −3 −1 −2 L2 − 2L1 5 −1 a 6 0 −6 a − 10 −4 L3 − 5L1 1 1 2 2 −1 −2 ∼ 0 −3 0 0 a−8 0 L3 − 2L2 ¤   (   $$  ©    ¨ E 9 H   ( )   $ ©    ¨     09$   © ( 0  (  )$ #  0 )   D D   $$ © 1 ( ' & % $ # 4  ¨  0  D D ¨ 0   0  C X  0   $  0 0 P 4  ¨ 0   $  0 ¨   ¨$  B ) E 0    ¨  a=8 a=8 x1 +x2 +2x3 = 2 2x1 −x2 +3x3 = 2 5x1 −x2 +ax3 = 6.  D D ¨ 0   0  C X   E      0 F 1 !       £¤ ¤ ¥  ¦ ¨  ¢  ¡ £ ¢ ¢ ¥ ¤ ¨  ¢ ¤ ¢ £  ¦ ¨ £ ¤ ¢ ¨ £ ¢ £ ¥ ¡ ¢ ¨ §¦ ¥ ¤ ¢£ ¢ ¡ 'Q 4 Q  0 9 $   %  0 9 $  3 $   C ©  X   ¨ $    ¨  ©  B   0  ) $ #  9 0  0 9 $  X   H     H     ¨©  ©©¨ § 4 0  $   ¨$ " X$      C  E H 9 4 4 4 ¨4 H6  ¨ 0 ¨ G 0    $   0 ¨  D D ¨ 0   0  C X    ¨  E H ¨    E X1 b3 − 2b2 − b1 = 0 b3 = b1 + 2b2 A 1 1 1 1 1 b1 2 −2 4 b2 ; 0 −4 5 −3 9 b3 0 −8 1 1 0 −4 ; 0 0 1 b1 2 b2 − 2b1 0 b3 − 2b2 − b1 1 b1 2 b2 − 2b1 4 b3 − 5b1 b3 − 2b2 − b1 = 0 L3 − 2L2 L2 − 2L1 L3 − 5L1  )$ #  )   D D    X©¨ 1 4 0  $ $© V     C   I #   $ ©  $$ # D D        H  (   0  $ C ©    #$ )      U 0    () 1 4 0    $  0 ¨   ¨$  B ) E 0 I   D D¨ 0  0 C X   0 0  H  0 D D ¨ 0  0 C X $    B  9  (   0 $ C ©  X$ ¨   ©  B   ) ) ¨  ¨ 9 § 4 0   $  0 ¨    ! I   D D ¨ 0   0  C X    0 0  H 0 0     X$¨   ) ($© U  ¨    C 0 ( ©  B )   1 4  ¨ 9 H  (  ¨ ©   ¨  0   0 0 b1 , b2 b1 , b2 b3 b3 rank(A) = 2 Ax = b b # 111 11 2 −2 4 ; 0 −4 5 −3 9 0 −8 11 0 −4 ; 00 1 2 4 1 2 0 L3 − 2L2 L2 − 2L1 L3 − 5L1 ¤   0     ¨     0 9 $ U 0   X ¨ ¥ 4  0 0  D (   X$ ¨ ©  (  ($$  © ¨ E 0 )$ #   ()   $ ©  ¨    09$  © ( 0   9 H  ¨  © ¨   ¨   § 4 U0  0 ) ¨ W 4 0   $  0 ¨   ¨$  B ) E 0 ©    ) ) ¨  0   ©    D D ¨ 0   0  C X  ©¨ E H  ¨ 0 ¨ 0    $  0 ¨    !   ©$ $   ¨  E H ¨     © ) #$ E    ©¨  ¨ H  2 4  0  3$ C X)$ E 3 ©0 X  ¨ E H    0  ¨  0 ¨   # 0    $  0 ¨    !    ¨  ¨    E X E H 96 Q Q4   0 0 ©¨ 1 E   (    1 4 0    $  0 ¨   ¨$  B ) E 0   ©$$  I    D D ¨ 0   0  C X    (  ©$  0   X  ¨   H    3$  E H RQ4   0 0 © ¨ 1 ©  B )   1 4  )$ #  0 0  ¨   ¨        © 2 rank(A) Ax = b Ax = b rank(A) rank(A) < 3 Ax = b $ Ax = b $ b rank(A) = 3 A rank(A) = n 111 A = 2 −2 4 . 5 −3 9   X©¨ 1 ( ' & %$ # $ Q 4  )$ #  0 0  ¨    ¨  (    ¨   ©    ¨ H  ) $ ©  ( © © E  0    D  E © © ¨      ¨  )$ #   0 0  ¨    ¨  (    ¨   ©    ¨ H  ) $ 3 4  B ©   C  $$ C  (  )$ #  0 0¨ $$ C  ¨ U X H 0 )$ #   ©0 F U0   C ¥ 4   (  G E  0 0  3$ C X)$ E 3  I ¨ ( 1 4   (  E 0 0 3$ C X 09$  I ¨  )$ #P 4 (  G  3$ C X 09$ 0 X   ¨    3$ C X)$E 3    ¨  0 ) ¨    V ©  B    $ 3 4  )$ # V   (  )$ # V  G  )$ # V      X ¨ 1 4  )$ #   0 0  ¨    B !   ¨  (    ¨ H  ) $ © (     () 1 4  )$ # )9$  ¨      C  0   ( ©  B )      3$   E § 4  )$ #   V    B !     X ¨  G   B ©    C         (    ¨ 9 § 4  3$ C X 09$ 0 X  ¨    3$ C X)$ E 3    ¨ E H  3$ ¨     0 )¨   V ©  ¨$   C     ¨ H $©     E X1 4  )$ #  0 0¨  ¨  (   B ! ¨ ©¨     © #    () 1 4  )$ #   0 0  ¨    ¨   H     ©  ¨  )$ # C ©       9 H  © ¨ ) $ © 9 D D    © D ) 1 4  )$ # © U ©  B   )$ #  ¨  (  )$ # ¨ ¨  H ¨   D D    X W 4    ¨  0 I B    ©     ¨ §   ( ' & % $ # 4 ¨ ¨ E H  1   4  ) $ #    0  0  ¨   X   H   H  ¨ E H  ) $ #    0  0  ¨    ¨  0 0  X) $ #  C ©  ( 1  4  #   () ¨  0   0 0 ) ¨© U   B )      V ©B     0 0  H V   0 0  X) $ #    B ©       #   E   ¨ ©  ¨ Y 4  ) $ #   (  0 ) E ©   ©  ¨ E  V ¥ 4   0 0   "   ¨   0 0     ¨    0   #$$    3 0      © ¨ ©  (    $ ©   ¨ 1 ! 2 ¡     m×m A m=n A A B C C C C C B p=n C n×n A C B p×q B AB = C C q=m=n A B C m×k A A=O A n×n AB A n×n B A AB m×n AB = C AB = O A, B k×n B B=O C 3×2 0·2+1·4 0·1+1·3 43 01 21 = 1 · 2 + 0 · 4 1 · 1 + 0 · 3 = 2 1 . CB = 1 0 43 2·2+3·4 2·1+3·3 23 16 11   0 ) ¨ W 4  )$ #  ¨ 0   ( ) © U (  G E    0 0  3$ C X)$ E 3  ( E 0 0  3$ C X 09$  I¨ 0  () © 4 ©0¨ $ ) ¨  3$ ¨    (  % ¨  3$ C X 09$  ( % ¨  3$ C X)$ E A 4  3$ ¨     X)$ #  0 ) ¨   0 U0    1   4 ©0 ¨ $ )  ) )¨  3$¨    X)$ # ¨ 9 § 4 R ¨  0 0¨  ¨ H  3$ C X 09$  ( % ¨  0 )$ #  #  3$ C X)$ E A 4  )$ #  ¨  (  )$ # ¨  ¨   H  3$ ¨     0 ) ¨     0  ¨ !    ¨ V ¥ 4  0 0¨  ¨ H  3$ C X 09$ 0 X ¨  0 )$ #   #  3$ C X)$ E 3 ¨  X)$ #  X  ¨ ©  3$ ¨     0 )¨  ©  B   ¨  0¨  2   B C C " CB BC B 2×2 B BC 3×2 C XY AB = 12 34 21 1·2+2·4 = 43 3·2+4·4 1·1+2·3 10 7 = . 3·1+4·3 22 15   0 )  ¨ W   ( ' & % $ # 94 8 2 17  U X ¨ Q4 Q4 %   B A6 4    (   G   G © 0  ¨   $  )   ) ) ¨  ¨  3 $ ¨       X) $ #   3 0   ( )    0  ¨   ¨     F )  © U   ¨ G   0 )  ¨ W AB BC CB 12 34 A= 01 C = 1 0 . 23 21 43 B=   X©¨ 1 !  ¡     £ ¢ ¤ ¡ ¢ ¨ §¦ ¥ ¤ ¢£ ¢ ¡ 4Q 4 0(  V  ( 0 )$ #    ) )¨ 0 U  3$  4 34 © G  )$ # $$ C    ) ) ¨ ©0 0¨ $  0  B 3 9 0  0 X  3$  9 § ( ¨ E H (  ¨    B 3 $  3 4  X) $ #  0  3 $ C        © 0 0 ¨  $   ) ) ¨   3 $   0  $  ¨  $ ¨ 9  4  ¨  0 ¨    (  ¨  U0  E X1 B 10 02 B= AB = 11 22 BA = 12 12 11 11 . , −AB + BA = O  ¨ G A A= AB = BA (A + B )(A − B ) = A2 − B 2 (A + B )(A − B ) = A2 − AB + BA + B 2 .      E    0     U D D    3$ C          ¨ § ( ' & % $ # 9 8 P7  U R S4 R4 Q   B A6 4 © © V  V  0  0   X   $ ©     3    #$ )       0 0 I  (   B 3 © "   I ¨  E H  ) ) ¨  ¨ (   ¨ 0 ¨ G   H  0 0   E H  ¨ (   1  1 4  )$ #  0 0 ¨   ¨ 4 4  0 C  9 H  ¨ 0  0   #$$    (  0  B 3 9  0   0   #$$       ¨ H  3$ ¨         0 0  H G  )$ #$$ U 0    ¨ H ©    (  G (  ©E 0 ! 5 ¡       )9$    E X   3 0 #¨ W E ©   0  ©   © © ¨ § 4  )$ #$$ U 0  ¨  (  )$ #  3 0  9 H   C ¥ B B2 = O B 00 1·0+0·1 1·0+0·0 00 = O. = = 00 0·0+0·1 0·0+0·0 10 ( 10 00 4  ¡ 10 00 £ (A + B )(A − B ) = A2 − B 2 A A AB = ¨ E § A= B= 00 . 10   X©¨ 1  ¨  0    # $ $      ¨ §     E ( 5Q £ (A−1 A)x = A−1 b 9 H  ¨  0  D D ¨ 0  0  C X 0    0 4   ¨ ¨   0 0  0  C X 9   $     E !   3$ C     ` x1 1 = . x2 7  ¨   ©  ) 9$ E   D D ¨ 0   0  C X  0  ¨    ¥ 1 d −b −7 3 = . −5 2 ad − bc −c a 9 44 5   0  0 © ¨ 1 ©  B ©  ¨$ 0   ¨  3 0   )$ #   ¨  E  3$     ¥ 4 ©  ¨$ 0    ¨  3 0  V   )$ #    0 0  H  $ C ©   $$   0 0 P ! 2 £     1 14 = . 7 9 A−1 G 242 1 r 3 121 r 242 1 r 3 112 r x1 −7 3 = x2 −5 2 x = A−1 b Ax = b 2 −3 5 −7 A−1 = )  6 ¨ 0  ¨  3 0   ( ©  ¨$ 0 ¨  3 0  )$ #  ¨ 9 § ad − bc = 2 · (−7) − (−3) · 5 = 1 = 0. ¨   B 3 ¨ H A 4 ad − bc = 0   ©  ¨$ 0    ¨  3 0   0 ¨   9 H  ( E H A= ab cd  ¨  0 0 ¨  ©  B )   1 4   ©  ¨ $    #   U 44 5   0  0 © ¨ 1   ) ) (   © ( ) ( ' & % $ # 2x1 −3x2 = 1 5x1 −7x2 = 7.  D D ¨ 0   0  C X   # ¨$   $ ©  0   C ©     0  © ( ) 4  0 ) ¨   ( ©  ¨$ 0    ¨  3 0  V  A−1 A= 2 −3 5 −7  )$ #     # 0 ¨     ¨ Y !  £     £ ¢ ¤ ¡¢ ¢ ¢ ¥  ¦ ¢  ¡ ¢ ¨ §¦ ¥ ¤ ¢£ ¢ ¡ TQ ¤  0 ) ¨    (   © E 0 4 0   ( E  )$ #  )   E       )    9 H  ¨      ¨     0 9 $   © ¨ !  (  )$ # 0 )   D D  $$ ©   E   X #    ( ' & %$ # [A | I ] [I | C ] x+3y −z = 1 7x +z = 2 x +z = 3 . 1 3 −1 7 0 1 . 10 1   D D ¨ 0   0  C X    # ¨ 0    0    ) $ #     ¨  3 0    0 )  ¨ W   ! 5 £     4    ¨   3$          ©  ¨$ 0   ¨  3 0   ) ) ¨  )$ #  ( R 0 ¨  0 0   0 )$ #  0 0 ¨ 9 ) 0  4 ¨6 ©©V © ¨ 9 § 4  "& ¨    ! ¨  ¨ 09$   0¨   C ¥ r 2 4 2 242 1 r 3 ; 0 r − 2 2 L2 − 1 L1 . 2 1 0 0 0 L3 − 2 L1 121 r=0    !  0   0 0 D   ¨ E H ¨ 0¨ G r=0 ¤  )$ #  0 0 ¨    H  ¨ ( 1 4 ©  ¨$ 0   ¨  3 0   ) ) ¨  )$ #  (  ¨ © ¨  )$ # ©  ¨$ 0  ¨  3 0   ( G  0 0 D E X H   C ¥ 242 2 4 2 1 r 3 ; 0 r − 2 2 L2 − 1 L1 2 112 0 −1 1 L3 − 1 L1 2 2 4 2 ; 0 −1 1 L3 0 r − 2 2 L2 242 ; 0 −1 1 . 0 0 r L3 + (r − 2)L2 0 )$ #   # E   X #  4 ©  ¨$ 0    ¨  3 0   ) ) ¨  )$ #  ¨ E H 9  H 0 ¨    B    ¨  0   0 0 D  ¨ 0 ¨ ©  ¨$ 0   ¨  3 0   )$ # ¨ E H  0 0 D E X H   E   1 4  ($$  © ¨ E  I #  ¨ H    ()  (    ¨    09$  ¨  0 )$ # E 9 H ©    E    4    0 ¨   9 H  ( E H ©  ¨$ 0   ¨  3 0  V   )$ #     ¨   § 4   © ¨$ #  U S4 5   0 0©¨ 1   ) ) (   © ( ) ( ' & %$ # rank(A) = n n×n A &% 4    ¨ 0   0    # $ $  P   ( ' & % $ # 94 8 P7 9 R %4 '4 Q   B 3  6 4  ) $ #       ¨  ) $ #  ©  ¨ $ 0    ¨  3 0  ©  ¨   V 1   4 ©  ¨$ 0  "   ¨  3 0   ) 9 $   ) $ #   X   H  ¨ E H  ¨ $ 0    ¨  3 0   ¨  0 0  X) $ #  C ©  (  1   4  ¨ E H G ©  ¨$ 0   ¨  3 0   ¨  (  1   4 8 0 ¨   )  ¨    0    0    #$ $      0 0  H   B ©  8  © © V     V       #   E     ¨    ¨  G  (  ) $ #     #    ( )    0    #$ $    3 0      ©  ¨ A ! @ £     4  0 0C X  ¨ 7  D D  9 0 0  0 0  0    $   0 #©    9 H  ¨    © © V    ¨    0  0 ) ¨  © U    0 # ¨  0 0      ¨ ©$ $ ¨   2 C A=B " AB = C AC = BC A, B C 1 1 x 0 −1 1 −6 6 6 4 y = x = A−1 b = 1 − 1 = 13 2 3 9 9 9 7 19 z 0 −1 3 6 6 6 9 H ¨  0 D D¨ 0  0 C X 0   0 4 $"9 6 9 ¨    (   (  )$ #      ¨  )$ #  $   © 1 4  0   (  E  D D ¨ 0   0  C X   © W 4    0 0 ) ¨© U 9 $$  ¨   0 0  0 ¨ G   0 ) ¨ ©©V  ¨ E H  )$ #   0 0 ¨ ¨ 0   ()©U ¨ @  3$ ¨       0 ) ¨    9 H  ¨   © © V   V   0    0 0 ) ¨  © U   (    0 # ¨  0 0     Ax = b A I A−1 = 0 1 3 0 1 6 1 −9 −1 6 4 9 7 6 −1 6 AA−1 .    ¨  0   0 0 ) ¨  © U  U 0    ©   ) 1 3 −1 1 0 0 L3 10 1001 7 0 1 0 1 0 ∼ 7 0 1 0 1 0 10 1001 1 3 −1 1 0 0 L1 10 100 1 L2 − 7L1 ∼ 0 0 −6 0 1 −7 0 3 −2 1 0 −1 L3 − L1 10 100 1 0 3 −2 1 0 −1 L3 ∼ 0 0 −6 0 1 −7 L2 10 10 0 1 0 3 −2 1 ∼ 0 −1 1 7 00 1 0 −6 − 1 L3 6 6 1 1 L1 − L3 1000 −6 6 4 0 3 0 1 −1 L2 + 2L3 ∼ 3 3 7 1 0 0 1 0 −6 6 1 −1 1000 6 6 4 1 L ∼ 0 1 0 1 −1 3 9 9 32 1 7 0 0 1 0 −6 6 Q% 4  ¨    09$  0 0 ¨  0 ¨    ¨   0 )$ #    0 0 ¨  U $ ©   H  U !   ©  B   ¨  ) ) ¨ 9 H G 9 ¨ ©  #  ©0 ¨  ¨$¨ 4 ¨6  )$ #    ) )¨  ¨   H 0 ¨ G ©  ¨$ 0 ¨  3 0  ¨  )$ #P ( 0 )$ #    () 1 4    ¨ 0   0    # $ $ P   4  0 ¨ $ !    V   ¨  )$ #  ¨$ 0    ¨  3 0    ¨  (    ¨ H $$ ¨ $ 3 4 ©  ¨$ 0   ¨  3 0  E H  ¨ '4 5   0  0 © ¨ 1 ©  B )     (  ¨ $ 0    ¨  3 0    B !   ¨  ( 0  " ) $ #P 4  ¨ E § 4  )$ #  ¨$ 0   ¨  3 0   V  (    #  E  U0    ¨ Y 4 ©  ¨$ " 0    ¨  3 0   ¨ ( 1 4 ©  ¨ $ 0    ¨  3 0    3 $ ¨       ¨ E H   ©  ¨ $    #   U '4 5   0  0 © ¨  ©  B )  1 4  ¨ $ 0    ¨  3 0   V   ( 0  ) $ #     )  ¨ $   C  0 0  H      ( )  ©   #P 4    ¨ 0   0    # $ $  P   4 ¨ 9 H (  ¨ 0  9 0C X E 14  ¨    $     E !   3$ C       (  0 0C X   ) C 3 I= 10 01 A 21 11 A−1 = 1 −1 −1 2 . C B = A−1 C B = A−1 C C C C A AC = BC = (BC )C −1 CC −1 B C −1 A=B =I   $ ©    B  E   A= B A−1 A AB (AC )C −1 %% Ly = b 3 1 y1 y1 0 = 1 y2 = . y2 1 111 y3 y1 + y2 + y3 ¨ E § 4  0  0  C X     # ¨$ E H  0 0 I   $ 3 4   X©¨  ( 0  ( E  0 0 C X  © W 4 S4 T ¨   X  $ #P 4   D D ¨ 0   0  C X   # ¨ $   $  ©  0  0  © © E H "  © ( 0    ¨ 0   9 1 y = Ux 1 11 1 A = 1 1 1 . 111 −1 "    0 ) ¨ W LU x = b y 1 1 1 = 1 1 = 1 . 1 1 11 111 1 ( LU ¨ 0 ©©E H ( 1 A 1 1 1 − E2 1 = 1 . 11 1  0 0  X) $ #          ¨  3 0  © U  0 )  ¨     ¨ © $ $ ¨   2 4 LU − − L = E1 1 E2 1 − E1 1 = ¨ 9 §     E    ( ¨ ( 1 − − L = (E2 E1 )−1 = E1 1 E2 1 U 11 U = 1 1 1 . −1 (   ) ) (  "  ¨    (   $$  ©    1 4 %  ¨  )  9 U 0    C P 111 1 0 1 1 ; 0 121 0 1 0 ; 0 11 1 1 0 −1 11 1 1 10 L3 − L2 L3 − L1 1 E1 = −1 1 1 1 E2 = −1 1 1 ¤ ©U 0(    ( )  © B   © © ¨ § 4  ©     ¨   09 $  3$         ¨   0 9 $ E  $ ¨9    ¨   0 0  !       9  I  ¥ 4  )$ #      3 0         (   0    ¨     09$  I # 3$   ) " !  3$     ©     0   X  (   (   $ $  ©     ¨ E  0  ) $ #    ( )    ¨  ¨  )   ©   #P 4 %4 T   ¨    0 )  ¨ W   X  $ #P ( ' & % $ # 3 1 1 1 x1 0 1 1 x2 = 0 . A= 1 1 2 1 x3  0  0  C X   # ¨$   $ ©  0 0  © © E H "  © ( ) LU 111 A = 0 1 1 . 121  0  )$ # 0  © © E H "  0 0 P !  ¤     LU  £ ¢ ¤ ¡¢ §¤ ¡ £   ¨ § §§ ¢ ¢ ¨ §¦ ¥ ¤ ¢£ ¢ ¡ ¡¡ R% 4 ©  ¨     E)   ¨   © V  $    ( ©  ¨      ¨   © V  © ¨ 1       ¨   ¨$ " X$   2 4    ( 1 C = 2 (A − AT ) C A = B+C $    ( 1 B = 2 (A + AT ) B C T = (A − AT )T = AT − (AT )T = AT − A = −(A − AT = −C. ¨ E § 4 4 © ¨      ¨ U0   X©¨ 1       0 ¨  ¨ E § B = A + AT C = A − AT B T = (A + AT )T = AT + (AT )T = AT + A = A + AT = B.     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 2      ¢ ¢¢  ¢ ¥ ¢ £   ¨ ¢ © ¢ ¨ § ¦ ¥ ¤ ¢£ ¢ ¡ &R 4   U  0  $  #  0 0   9  0    )$ E X   ) ) (  ¨   D D ¨ 0  0C X    $  "   #¨$   $©  © ( 0   C    ¨ 0 ¨        E X    9   B 3  ©©¨ H ¨ ` 4 R ¨  0  U  0  $ 3 39  (    E X H  ¨    U  0  $  #  0 0  Y 4   0 0 0 C X  U   0   $   # 0 0   9 H  3 0 #   0      4  "  E  "  V   9  H ©  0            © ) # $ E R4 Q Q   0  0  ¨   $  )  ©  B )   1 4 (     ¨  © 0  4 S  ( 4 R  (   ©  V   0 0  0  C X 9   © 0    0 0  E ©   (   ¨    ¨ § 4   E  "   ¨$  09$  0       ¨ E H  ¨  0  C X       ¨ H 0    $  0 ¨  1 c2 = 0  0 0 C X    () 1 4 R4 Q Q   0 0 ¨ $ ) 1 9 ©0 ¨ !  9 ¨   ©¨ ! 0 0¨ 0 ¨ 44 Q Q  ¨   0 )¨ W  © ( 0 ©© B  4 34 © G  E  "   ¨$ 09$ ¨ 9 H  0     E X  ¨ ©$$ ¨   2 4   U  0  $  0 0  D      ¨ H  (  0 0   © )¨ ©0       ¨$ 09$  ¨  ) 9 H E  0  $  X¨ V 1 (−1, 1, 0, 0), (−1, 0, 1, 0) G (−1, 1, 0, 0), (−1, 0, 1, 0), (−1, 0, 0, 1) c3 = 0 c1 = c2 = c3 = 0 c1 = 0 −1 −1 −1 0 1 0 0 0 c1 + c2 + c3 = . 0 1 0 0 0 0 1 0 (−1, 0, 0, 1) x1 = −x2 − x3 − x4 −1 −1 −1 −x4 −x3 −x2 −x2 − x3 − x4 x1 0 0 1 x2 0 0 x2 x2 = = 0 + 1 + 0 = x2 0 +x3 1 +x4 0 . x3 x3 1 0 0 x4 0 0 x4 x4 ¨ 9 § 4 ¨ ¥ 4 x1 + x2 + x3 + x4 = 0    E X1 4   0 0 0 C X 0  $    0   ) 1 ( ' & %$ # £  0   U   0    $ 3 39  ¨   0 0  0  C X  U    0   $   # 0 0     0 0 P ! @ 2     {(1, 2, 3, 4), (4, 3, 2, 1), (1, 0, 0, 0), (0, 1, 0, 0). 9 H  ¨    ¨   ¨ !  ¨  0 0 0 0  Y 4   # 0 0   9 H 3 0 #    )$ E 3  ¨ H 9  0     ( 0  ($$  ©  ¨ 9  0  )$ E 3   " X  © # 9   0 0 D  ¨   § R4 1 2 3 4 4 3 2 1 1 0 0 0 0 1 0 0 0 0 1 0 14 1 0 0 0 0 −5 −2 1 0 0 ; · · · ; 0 0 1 −2 1 0 00 0 7 −2 1 0 0 . 0 1   (   $$  ©    ¨ E  0 )$ #   ()   $ ©    ¨     09$ U 0   © ( ) 4 )$ E 3 E H 9   0 0 D   ( )   H          ¨   E  "   ¨$  09$   ¨   0  )$ E 3  ¨ ©  ©   # 9  0      ¨    § 4 %Q4 %Q   ¨     0 ) ¨ W  © ( 0   9 H 9 0    9 ©  $ ¨  RQ4 %Q   ¨     0 ) ¨ W 9 R  ( %  ¨  ) 1 4 ©$$   0 0  D  9  0     )$ E A A R4 1 2 A= 3 4 4 3 2 1 1 0 0 0 0 1 0 0 0 0 1 0 0 0 . 0 1  )$ #   E P 4  0      ( ) 0  9   (   0  )$ E 3  ¨ ©  ©   # 9   ( )  ) ) (  0      3$   E D D 4 Q I ¨  ) 1 9  0 0 ©  )   (  ©  B )     )$ # $ ©   U  4 RQ4 %Q   ¨     0 ) ¨ W   © ( ) ( ' & % $ # 4 (  0      3$ ¨  0 0  ¨    # 0 0     0 0 P ! 5 2     4   # 0 0    3 0 #  (  0     4 0 0     3 0 #    U  © $   0 0  D   ¨    0       ¨ ©  ©  # (  )$ E 3 4 %  ( 4 Q 9   0 0 D   ¨   § e1 , e2 , e3 , e4 (1, 2, 3, 4) R4 (1, 2, 3, 4) (4, 3, 2, 1) (4, 3, 2, 1) W QR 4   #   0 0    ©     0   X  ¨  ¨   0     0 0  D  ¨    E X1 ¨ 9 § ( 0  0  4 0   (    V    3 0    ¨     ( $$  ©  ¨  0  $  (   C  B H   ¨$  B ) E 0  ¨  0   0 (−2, 1, 1) x2 − x3 = 0.    ) ) (  ¨  3 3  ¨ ¨  H G N (A) N (A) (x1 , x2 , x3 ) = (−2x3 , x3 , x3 ) = x3 (−2, 1, 1). x1 + 2x3 = 0  Ax = 0 111 102 0 −1 1 ; 0 −1 1 000 000 10 2 ; 0 1 −1 00 0 Rx = 0 Rx = 0 R −L2 L1 + L2 ¤ © " !  9    ¨     B ©    !      § 4   #   (   $$  ©     ¨ © ©     0 0 I  (   0    ¨     09$  ¨   I (    E    ¨    H    E   3$ C ¥ 4   # ¨$   9 H  ¨  ©  0 0 I  U $$ U 0   #   0 0   Y 4  ¨   U  )$ E 3   #   0 0   Y 4 )$ E 3 4 %  ( 4 Q 9  0 0 D    E   V ¥ 4   U  )$ E 3   # 0 0     3 0 #     (  $$  ©     ¨ 9  0 0 D  ¨    3 0 ¨   ¨    H  0 )$ #   ¨ 7   D D   U  )$ E 3  ¨ § 4   #  0 0    ¨   0 0  H G   U   09$   #   0 0      ¨ &    !  ) ) ¨  3$   0 0  ¨       09 0 A Ax = 0 (2, 4, 3), (1, 3, 3) A {(1, 1, 1), (0, −1, 1)} R(A) R(A) 213 213 4 3 5 ; 4 3 5 1 111 333 L 33 111 L3 ; 4 3 5 213 L1 111 0 −1 1 L2 − 4L1 ; 0 −1 1 L3 − 2L1 111 0 −1 1 ; 000 L3 − L2 ¤   ¨ $    ¨ H E    © © ¨    ¨  © B     0 0 )  ¨  © 4  #  ($$  © ¨  0 0 I   E   X #  4 S4 RQ  ¨   0 )¨ W   X$ #P ( ' & %$ # 94 8 2 17  U !Q 4 S4 R   B A 6 4   0 0  E ©  (  © © V  0  (    ¨  U  © $       (  0 0   U  © $   3 39  ¨   X¨ ¥ 4 ( 0  U  © $    #  0 0     0 0 P " A R(A), C (A), N (A) 213 A = 4 3 5 . 333 N (AT )  ¢ ¢¢ §¨¤ ¦ ©   X©¨ 1 !  5     ¨ ¡  §¦ ¢£ ¢  ¨ ¢ ¨ ¢ ¡ ¢ ¨ §¦ ¥ ¤ ¢£ ¢ ¡ %R 1234 1 2 3 4 1 1 1 1 ; 0 −1 −2 −3 L2 − L1 3579 0 −1 −2 −3 L3 − 3L1 1 2 3 4 0 −1 −2 −3 ; 0 0 0 0 L3 − L2 ¤   E   ( )  ¨   H 9 ©     E W 4   (   $$  ©    ¨ E  I #   ( )   $ ©    ¨     09$  © ( 0   9 H  ¨  0      0 0$$ ) # 0 ( ' & % $ # 4  0 )$ # 9 #©$$ 0 4 ¨6 3 39 $$ U0  ( 9 ) 0  4 ¨6 3 39 3 0 #   0 ) ¨ W A 1234 A = 1 1 1 1 . 3579 A   0  ¨   $  ) 1 94 R & & %    ( 9 I "  D  6 ! 5 5      4 9 H  ¨   U  )$ E 3   #   0 0   Y 4  ) $ E 3 4 R  ( 4 Q 9   0 0  D   ¨  V ¥ 4  0    ( $  $  ©     ¨ 9  0 0 D  ¨   3 0 ¨  ¨   0 )$ #   ¨ 7   D D   U )$ E 3 E H )  ©   9 H  ¨  ©  0 0 I   U  )$ E 3   #   0 0   Y 4 9 H  ¨   U   09$   #   0 0   Y 4 &    !  ) ) ¨  3$   0 0  ¨   0 H  ¨  ©  0 0 I   U   09$   #   0 0   Y " (2, 1, 4), (3, 1, 5) "   (  $$  ©     ¨  U   09$  B H )  ©   9 (1, 1, 1, 1), (0, 0, 1, 2) R(A) 2234 1 1 1 1 1 ; 2 4456 4 1 0 ; 0 1 0 ; 0 111 2 3 4 456 111 0 1 2 012 111 0 1 2 000 L3 − L2 L2 − 2L1 L3 − 4L1 L2 L1 ¤  0 )$ # 9 ©        Y 4   (   $$  ©  ¨ E  I # 0 )$ #   ()   ¨  I¨ )  © #P 4 S4 RQ ¨   0 ) ¨ W   X$ #P ( ' & %$ # 2234 A = 1 1 1 1 . 4456  0  )$ #  U  )$ E 3  ( "  09$   #  0 0     0 0 P 9 R & & % ©  U  E 9 I "  D  6 ! 2 5     4 9  0 0   (   U$$ # ©©V 0 (    H ¨ &Q4 RQ   0 0 ©¨1 ©  B )  6 E ©©V 0 (  ¨  ( 9   0 0   (   U$$ # ©©V 0 (    H  ¨ &Q4 RQ   0 0©¨ 1 ©  B )  6 E ©© V 0  (   ¨   0  U  © $   #  0    0 0   E   (    9 H  ¨  © V  0 0 1 4 '4 R Q   0  0 © ¨ 1     B       "  9   ©      ¨  ¨  " C (A) R(A) N (AT ) N (A) dim R(A) = 2, dim C (A) = 2, dim N (A) = 1, dim N (AT ) = 1,   ¨   0   U  © $  E   0  3 39  4   # 0 0   9 H  30#  0 0     4  ¨  0  D D ¨ 0   0  C X 0    0 4  D D ¨ 0  0 C X   # ¨$   9 H  ¨   0 0 I   ©  3$   0 ¨  ¨   # 0 0   Y N (AT ) (x1 , x2 , x3 ) = 3 x3 ( 2 , − 3 , 1) 2 3 ( 2 , − 3 , 1) 2 N (AT ) AT x = 0 RR 4 ( 9 ¨ 0 0    ( 9 ¨ 0 0    4  ¨ 0 0   0 4  0 0 C X   #¨$ 4 4 4 ¨4 H G  0 0    © ) ¨ © 0       ¨$  09$  ¨  0 0    U 0   © W 4    #    0 0    9 H ©  B     #    0  0 0        © B !  (    #  0 0  0 0      ) C ©      ¨ §   4  # 0 0    9 H  3 0 #  (  0      4  0   ©       $$    ¨  0  $     E   (  $  ¨    0 0  © 0 U ) ©  B )     D D ¨ 0  0 C X  ©© ¨ H   # ¨ 0 (x1 , x2 , x3 , x4 ) = x3 (1, −2, 1, 0) + x4 (2, −3, 0, 1) (1, −2, 1, 0) w4 = (2, −3, 0, 1) V⊥ V V⊥ R4 (−4, 0, 2, 2) w1 , w2 , w 3 , w4 c1 w 1 + c2 w 2 + c3 w 3 + c4 w 4 = b (c1 , c2 , c3 , c4 ) = (−4, 0, 2, 2) v = c1 w1 + c2 w2 = (−3, −1, 1, 3) v⊥ = c3 w3 + c4 w4 = (−1, 1, 1, −1) V⊥ b = v + v⊥ V w3 = 0 =w1 · x = x1 + 2x2 + 3x3 + 4x4 0 =w2 · x = 4x1 + 3x2 + 2x3 + x4 .  0  D D ¨ 0   0  C X  U   0    $ 9 H  ¨   U  © $ ¥ 4  ( E   B !  © © V  0  (    ¨  ¨       ¨ H   ¨$  B ) E 0   ¨ A 4   #  0 0    ¨ 9 § 4   E  "   ¨  (  0      (  0   (  E  ©  E  9    0  ¨   V 1 4 (   ©  W   ( ' & % $ # 4  U      (  U           ¨  0 0        ©  W   4    #  0 0      0 0  P   4 9  U  © $    ¨   ©E 0 ! @ 5     4 % )9$  ¨ 0 3 39 $$ U0 (  $  © 3 0 ¨ 0 E )$ E 3  3$ C X 0 C X  ¨ 0 3 39 $$ U0  ( G % ¨ 0 3 39 3 0 #  4 4 4 ¨4 H G $  © 3 0 ¨  3$ C X 0 C X  ¨ 3 39 3 0 # ` 4  $   ©  3 0 ¨ 0 E   ¨ S  ( R  )$ E A 4 )$ E 3 4 %  ( 4 Q 9   ( )  ¨    $   © 3 0 ¨  ¨ ©   ¨   § V w1 = (1, 2, 3, 4) w2 = (4, 3, 2, 1) c1 w 1 + c2 w 2 w1 w2 V⊥ R4 V⊥ V = Span((1, 2, 3, 4), (4, 3, 2, 1)) V⊥ b = (4, 0, 2, 2) V V {w 1 , w 2 } w1 w2 V⊥ R4 A ...
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This note was uploaded on 10/25/2011 for the course VéLAVERKF STÆ107G taught by Professor Ragnarsigurðsson during the Fall '11 term at Uni. Iceland.

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