¼±Çüº¯È¯À̶õ?
( ½Ã°¢È Âü°í ÀÚ·á´Â http://matrix.skku.ac.kr/sglee/LT/index.htm )
¼º´ë ÀÌ»ó±¸ ±³¼ö (TA : Á¤Áø, ±è´ö¼±¼±»ý)
4Àå ¼±Çüº¯È¯ (L.T)
(4.3, 4.5, 4.8 ÀýÀº ¼±ÅÃÀÔ´Ï´Ù.)
Goal : ¼±Çüº¯È¯ÀÇ MatrixÀ» ±¸ÇÏ´Â °Í, ±âÀú º¯È¯ÇÏ´Â °Í (Matrices of L.T's, Change of basis)
ÇÔ ¼ö (transformation, mapping)´Â ¶ó°í
Ç¥ÇöÇÑ´Ù. ±×¸®°í
Àº ½Ç¼öüÀÌ´Ù
º¤ÅͰø°£
1)
2)
Áï, À̰ÍÀ» ÇÑ Á¶°ÇÀ¸·Î Ç¥ÇöÇϸé
|
Áï, º¤ÅͰø°£¿¡¼ ¡° ¿¡¼
¹æÇâÀ¸·ÎÀÇ line À»
¿¡¼
¹æÇâÀ¸·ÎÀÇ lineÀ¸·Î º¸³»´Â ÇÔ¼ö¡± ¶ó´Â °ÍÀÌ´Ù.
º¤ÅͰø°£ V ¿Í W À§ÀÇ ÇÔ¼ö
¶ó°í Á¤ÀÇÇÑ´Ù. ( |
1)
2) |
[º¤ÅͰø°£°ú ¼±Çüº¯È¯(Linear Tranformation)°úÀÇ °ü°è]
Let
¢¤!
L.T. |
Pf) Let for some
's.
Define by
(ƯÈ÷
)
[Show T °¡ L.T. ] Do!
[Show T °¡ unique] Do! ¡á
Note
Let ÀÌ
Â÷¿ø vector spaceÀÇ "ordered basis"¶ó ÇÏÀÚ.
Let ÀÌ "standard basis" for
.
¢¡ ¢¤ isomorphism by
¢£
À̸¦ ¡°the natural isomorphism
w.r.t " ¶ó ÇÑ´Ù.
[ÀÓÀÇÀÇ ordered basis¸¦ standard basis·Î transferÇØ ÁÖ´Â Natural isomorphismÀÇ °³È²]
Let ÀÏ ¶§ ,
¸¦ the coordinate vector of
w.r.t a basis
, Áï
¶ó°í ¾´´Ù.
ÀÇ image
À»
ÀÇ coordinate vector w.r.t a basis
¶ó ÇÑ´Ù.
(i.e,
¶ó°í ¾´´Ù.)
º¸´Ù½ÃÇÇ natural isomorphismÀ» ÀÌ¿ëÇÒ °æ¿ì ÁÂÇ¥º¤ÅͰ¡ ÀÏÄ¡ÇÔÀ» º¼ ¼ö ÀÖ´Ù.
LT ÀÇ ¿¹
Identity transformation :
Zero transformation
:
Rotation on
Projection on x Ãà
Reflection about x Ãà
Trace of
¹ÌºÐ ÇÔ¼ö
ÀûºÐ ÇÔ¼ö |
Prob 4. 1 (±³Àç p. 123)
Prob 4. 2 (±³Àç p. 123)
Exercise À§ÀÇ Á¤ÀÇ Áß
ÀÌÁ¦ À§ÀÇ
Prob 4. 3 (±³Àç p. 124) ¢£
Prob 4. 4 (±³Àç p. 124)
¸¸ÀÏ ±×·¸´Ù¸é
Prob 4. 5 (±³Àç p. 124) Let L.T ¸¸ÀÏ (Áï , L.I. setÀÇ inverse image ´Â L.I. ÀÌ´Ù. ) |
4.2 Invertible L.T.
L.T. T: V -> W °¡ ¾Æ·¡ÀÇ µÎ Á¶°ÇÀ» ¸¸Á·Çϸé, isomorphismÀÌ µÈ´Ù.
1) 2) |
6) |
Pf) °¡ invertible À̸é
µµ invertibleÀ̰í L.T.¸¸ º¸ÀÌ¸é µÈ´Ù.
¿ì¼± °¡ (1-1 À̸é¼) onto À̹ǷÎ
¢¡ ¢£ and scalar k,
¢¤
s.t.
,
¢¡
¡á
Suppose
1)
2)
3)
4) If |
(°°Àº ü(field)
»óÀÇ) µÎ V.S. |
Pf) (=>) °¡ iso.À̰í
ÀÌ
ÀÇ basis À̸é
[Show ÀÌ
ÀÇ basis] (done!)
1) L.I. :
(¡ñ
°¡ 1-1 )
(¡ñ
µéÀÌ L.I.)
2) Span : ¢£ , ¢¤
So
µû¶ó¼
(<=) À̸é
Let (,
À»
¿Í
ÀÇ ±âÀú¶ó ÇÏÀÚ.)
±×·¯¸é ¢¤L.T. s.t.
¢£
[Show T °¡ 1-1 ] Do! easy
[Show T°¡ onto ] Do! easy
T ´Â isomorphism
¡á
L.T.
(1)
(2) If |
Note
¨ê T°¡ onto
: prob 4.7 (2)¿Í °ü°èÀÖ½À´Ï´Ù.
(ü(Field) |
(easy to prove)
* ordered basis : : a basis (in order) for
.
* standard (ordered) basis :
* natural isomorphism from to
:
by
±×·¯¸é
À̰ÍÀº coordinate vector of w.r.t the basis
Àε¥
XÀÇ coordinate vectot w.r.t the basis ÀÎ
°ú ÀÏÄ¡ÇÑ´Ù.
Áï . ¶Ç
Prob 4. 8 Find the Matrix of reflection about line
e.g.
Prob.4. 9 Find the coordinate vector of
w.r.t the given ordered basis
(1) (2) |
Let
±³Àç p.70 ÀÇ remark(2) ¸¦ º¸¸é
ÀÌ A´Â À§ÀÇ parallelepiped¸¦
Let
|
L.T.
(Á÷Á¢ »ý°¢ÇØ º¸½Ã¿À!) |
4.3 ÀÀ¿ë Computer graphics (skip)
computer screen »óÀÇ ±×¸²ÀÇ animation°ú graphical display.
L A ¸¦
Ãà¼Ò, È®´ë, ȸÀü, reflection
Ãà
,
,
(skip)
Find the matrix |
4.4 Matrices of L.T's
Goal : Show ¢£ L.T. ,¢¤! matrix
s.t.
for all
Let L.T. with ordered basis
,
.
¢¡ ¢¤! 's s.t.
for all
¢¡ (Áï, coordinate vector of
w.r.t
)
Let
[Show |
Pf) Let
LHS
RHS
À§ÀÇ matrix ¸¦ ¡°
ÀÇ matrix representation(Çà·Ä Ç¥Çö) w.r.t basis
and
(¶Ç´Â ÀÇ associated Çà·Ä w.r.t basis
and
)"À̶ó Çϰí
¶ó ¾´´Ù.
Áï, Çà·ÄÇ¥ÇöÀº ¹æ¹ýÀ¸·Î ±¸ÇÏ¸é ½±´Ù.
Note
,
·Î ¾´´Ù. (¾Æ·¡´Â
³» ±âÈ£¹ý)
Note
ÀÏ ¶§
ÀÎ similar Çà·ÄÀÇ °³³äÀÌ ¿©±â¼ ³ª¿Â´Ù.
Let
Find |
Sol) .
.
Answer :
Let
Find |
Sol) since
Let |
´ä :
¹® 4.12 (±³Àç p.139) Let
(1)
(2) |
Sol) (1) (2)
(1) (2)
Prob 4 .13 Let
Let
Find |
Sol) ,
,
,
Answer :
Prob 4 .14 Let
ÀÌ ¶§ |
Sol) ¢¤ 's s.t.
¢¡
4 .5 LT ÀÇ º¤ÅͰø°£
Note
p. 143 ÀÇ ¹®Á¦ 4. 17 À» Do!!
4 .6 Change of basis
¥á, ¥â°¡ ÀÇ µÎ different basis¶ó ÇÏÀÚ.
ÇнÀ¸ñÇ¥ : »çÀÌÀÇ °ü°è
then Áï, ÃàÀ̵¿À¸·Î x'y'-ÁÂÇ¥Ãà¿¡¼ standard ellipse¸¦ ¸¸µé ¼ö ÀÖ¾ú´Ù. |
Def. In General ,
Let
Let
¢£
±×·¯¸é
À̰ÍÀº
Âü°í
( ie. ÀÌ |
Note
Prob 4 .18 (±³Àç p.146) Find |
Sol)
Q =
Answer : =
4.7 Similarity
Let L.T T:
Let
(1)
(2) ¢£
(3) |
(skip) Thm 4.13 Let L.T T : (³»¿ë ÷°¡ ÇÊ¿ä.)
Def 4 .6 Let A, B ¡ô A is similar to B if
¢¤ nonsingular |
Prob
if |
pf) ¢¤ nonsingular ¡õ
¢¡
Let L.T T:
Let
±×·¯¸é
Find |
Sol) ,
,
,
,
¢¡
¡Å
Thm 4 .15 (±³Àç P.150) Suppose
If |
¿¹ 4 .21 ( read )
Let D: diff. operator on
Find |
Sol)
¢¡
¢¡
¢¡ ÀÌ´Ù.
¢¡ ³ª¿Â´Ù .
Let L.T ¥á´Â standard basis
Find Show they are similar . |
Sol)
¢¡
Let
¡Å ¡á
Find the matrix reflection about the line y=x in the plane R^2 |
Sol) T: R^ ¡æR^
T=A
y=x¿¡ °üÇÑ ´ëĪÀ̵¿Àº
T=
À̹ǷÎ
Ç¥ÁØÇà·Ä A¸¦ ±¸Çϸé
T(e_
)=T=
, T(e_
)=T
=
À̹ǷÎ
A=,
T
=
=
´ä:
Show that, for any matrices A and B in |
Sol) Example 4.4¿¡¼
The transformation tr : was defined as the sum of diagonal entries
tr(A) =
for A=[]¡ô
(R),is called the trace.
µû¶ó¼, tr(AB)=
=
=tr(BA)
Is there a linear transformation |
Sol) ¸¸¾à¿¡ °¡ a linear transformation À̶ó¸é,
À» ¸¸Á·ÇØ¾Ö ÇÑ´Ù.
±×·¯³ª
Áï, À̹ǷÎ,
´Â a linear transformationÀÌ ¾Æ´Ï´Ù.
À̻󱸱³¼öÀÇ ÀÐ°í º¸´Â ¼öÇÐ ÀÚ·á½Ç (http://matrix.skku.ac.kr/sglee)
¨Ï 2003 Prof. S.G.Lee, Dept. of Math of SungKyunKwan University