( m is the length of columns of A ) spanned by the columns of A : . Select those columns of B that contain a pivot; say those are Bi

Transcription

1 Assgmet /MATH 47/Wte Due: Thusday Jauay The poblems to solve ae umbeed [] to [] below Fst some explaatoy otes Fdg a bass of the colum-space of a max ad povg that the colum ak (dmeso of the colum space) s the same as the ow ak (dmeso of the ow space) of ay max Rak of A = ow ak of A = colum ak of A Let A be a m max The colum space of A colsp( A) s the subspace of m ( m s the legth of colums of A ) spaed by the colums of A : colsp(a) = spa( A A ) whee A A ae the colums of A Oe way of fdg a bass fo colsp ( A) s do colum educto o A ; that s ow educto o A Aothe way s moe teestg howeve ow We ow-educe the max A A B B ow-educed Deote the colums of B by B B ; that s B = [ B ] B Select those colums of B that cota a pvot; say those ae B whee (Of couse the < < = ow ak of A ) The desed bass of colsp (A) s ( A A ) I wods: a bass of colsp( A) s obtaed by selectg those colums of the ogal max A that coespod to the pvoted colums of the ow-educed veso of A (Note that ths shows that dmcolsp( A ) = =dmowsp( B )=dmowsp( A ) as pomsed) Why s ths ue? The fst pot to obseve s that ( B ) s a bass of colsp ( B) Ths s easly see smlaly to the way oe sees how B gves se to a bass of ullsp( A) Secod pot: as we kow fo ay X A X = f ad oly f B X = () (ths s the bass fact of the equalty ullsp ( A) = ullsp ( B) ) Theefoe x A + + x A = f ad oly f x B + + x B = : ()

2 () just take X x x meag that each ey = [ ] x = x the vecto X fo whch s ot oe of the pvot-dces s zeo: Sce B ae lealy depedet x B + + x B = holds oly f x = = x ; but the by () the same holds fo A A place of = B B We just poved that A A ae lealy depedet We ca also see that ( A A ) s a spag set fo colsp( A) (theeby: a bass of colsp( A) ) ; fact a way that s useful fo calculatos Befly a abay colum A of A s expessed as a lea combato of A A the same way (wth the same coeffcets) as B s expessed tems of B Sce B s ow-educed to expess B as a lea combato of B s easy: t s smla to wtg the geeal soluto of B X = We ae sayg that the A s expessed as a lea combato of A A the same way Remde: ow ule ad colum ule fo max multplcato The colum ule of max multplcato says: fo maces A= [ A A ] : A s m B = [ B p ] : B s p (whee we have dsplayed the colums of both maces) we have ad each A X = AB = A [ B] = [ AB AB ]; p x j A j fo ay colum j= p X = [ x x ] I wods whe fomg the poduct A B we take a lea combato of the colums of A usg the ees of a colum of B as coeffcets to get a colum of A B Thee s a smla ow-ule: to get a ow of A B we take a lea combato of the ows of B usg the ees of a ow of A as coeffcets (Note the techage of A ad B the two ules)

3 3 Wtg a subspace U the fom U = ullsp( C) wth C a sutable max Suppose a subspace U of s gve the fom U = spa ( A A m ) Let us wte A fo the max A= [ A A ] m A s a m (!) max Ths makes U =spa ( A A ) = colsp( A) Cosde the max A ; A s a m max m Calculate a bass of ullsp(a ) the usual way; deote ths bass as ( C C k ) Let C be the max whose colums ae the bass vectos just foud: C = [ C C ] The C the aspose of C s the desed max: U = spa( A A m ) = ul lsp ( C ) Ca you pove ths? k 4 Sum ad tesecto of subspaces Fo the tesecto U W of two subspaces (of the same space ) see p 3 Lpschutz Fo the sum U + W of two subspaces see p 34 We have the mpotat fomula dm( U + W) = dmu + dmw dm( U W) stated as Theoem 4 p 34 ad poved solved poblem 48 o p 7 Geeal suctos: Use advaced max methods always You methods should be effcet fo poblems volvg lage maces Fo smalle maces such as the oes the poblem set hee oe could ofte use dect methods whch ae less effcet but would wok the example at had It s ot bad to y a dect method just to see what s gog o; but fo the aswe submtted as pat of the assgmet you should use the (bette) max method(s) May of the poblems come wth the suctos to use a specfc method o somethg pevously doe Sometmes othe pobably less effcet ways would also wok fo the poblem at had especally sce the poblem s usually made umecally easy Fo full maks (ths apples to the examatos too!) you should follow the suctos ad ot go the way of possble othe methods As a geeal emak: the couse ad the assgmets ae ot geaed towads solvg toy poblems but athe to leag ceta specfc methods of solvg possbly lage poblems oes whose complete hadlg eques usg a compute Havg sad all the above I have to say ths: you ae fee ad eve ecouaged to pot out (fo dscusso wth me o class) f ad whe you fd that some method poposed 3

4 hee (o ay othe assgmet) s ot optmal I am ot petedg that I fact kow the last wod about eveythg lea algeba [] Suppose that X X X X We let 3 4 Z = X X3 Z = X3+ X 4 Z3 = X3+ X4 Z4 = X + X3+ X4 Z = X + X 3 ) Use the method of the poof of the Ma Lemma class ad fd scalas y fo = 34 such that a) ot all y ae equal to ad b) yz = ) Assume addto to what we had so fa that the vectos X X X3 X 4 ae lealy depedet Pove that wth the Z s defed as befoe f we have two solutos Y ( y ) = ad Y = ( y ) of ) [that s thee s such that y ; thee s such that y ; yz = ; ad yz = ] the Y = a Y fo a uque scala = a [Wag: ths s ot a geeal fact; t holds ths specal umecal case] = = [] Let A be the max A 3 = 3 4 colsp 3 4 ) Fd a bass of ( A) ad wte each of the fve colums A A A A A of A as a lea combato of the bass vectos (Ht: cosult above) ) Use the wok fo ) ad wte the max A the fom A= B C B a sutable 4 k max C a k max wth the umbe k chose the least possble value (Ht: cosult ) 4

5 3) Use the wok fo ) to fd a bass of owsp( A) ad wte each ow of A as a lea combato of the bass (ow-)vectos foud (Ht: cosult ) [3] Let X = X = X 3 = vectos Let 3 a subspace of U X X X3 = spa ( ) ) Fd a k max C wth the smallest possble value fo k such that U = ullsp( C) (Ht: cosult 3) ) Usg the wok fo ) fd values fo the paametes a ad b such that the vecto Y = [ a b a b] belogs to the subspace U [4] ) Fd a bass ad the dmeso of U = spa ( X X X3 X4 X) subspace of whee a a+ X = X = a X3 = X4 = a+ 3 X = a+ 3 a a+ a+ a a a a a ( a s ay gve umbe; ts value s uspecfed) ) Fd a bass ad the dmeso of the subspaces V W ad V + W of 4 whee V = spa( X X X ) W = spa ( X X 3 X4) ad the X ( = 34) ae as befoe 3) Wthout futhe calculatos deteme a bass fo V W Justfy you aswe (Ht: cosult 4 )

6 [] We let X = X = X3 = Y = Y = Y3 = Y4 = vectos Let U = spa( X X X ) ad W = spa( Y Y Y3 Y4) subspaces of Deteme a bass ad the dmeso of the subspace U W of (Hts: Fd maces C ad D such that U = ullsp( C ) ad W = ullsp( D ) Let C A = the max whose ows ae those of followed by those of D C D The U W = ullsp( A ) (why s ths ue?))