IGCSE/GCSE/O & A Level/IB/University Student Forum
Qualification => Subject Doubts => IGCSE/ GCSE => Math => Topic started by: joel on May 17, 2009, 12:03:22 pm
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How do i find the matrix which represents a transformation
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i have the same problem refrence Q7b OCt08
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very simple....
first draw the unit matrix n den watevr matrix is given draw dat...n den chk wat kinda transformation is hppning between the unit matrix n the one given...
hope u get it.. :)
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zara can u plzz solve the Q7b in oct08
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first write down 2 sets of co-ordinates from triangle T in a matrix form
then multiply it with the unknown matrix and equate it to the matrix form of two corrseponding sets of co-ordinates of the image i.e R
(2 8 ) * (w y) (-4 -4)
(4 4) (x z) = (-2 -8)
therefore you can find the values of w,x,y and z by multiplying (-4 -4) with the inverse of (2 8 )
(-2 -8) (4 4)
=> so the answer is (0 -1)
(-1 0)
plz infrm me if u didn't understand...as i am available anytime in studentforums...
Good luck
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im still confused:(
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Your explanation is good!
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hmmm shan2391 c u can du it da way zara or eddie_adi619 told or in our skul dey hv thot us da base vectors which r matrixes n represent da transformations. da only bad part is dat u hv 2 lern dem. du u want dem?
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base vectors are life savers!!
and they dnt tke long to lern
5 minutes maximum...
gd fr my level ..
which isnt too high .. =P
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@sweetsh - thanx a lot :)
@angel - okk...so lets try the next sum
U onto Q
take any 2 co-ordinates of triangle U i.e = (-6,2) and (0,2)
arrange them vertically o form a 2 by 2 matrix
(-6 0)
(2 2)
now let say the unknown matrix of transformation is
(a b)
(c d)
so it implies that when you multiply the co-ordinates of U with the matrix of transformation, you get the co-ordinates of the image i.e triangle Q...(look at the corresponding co-ordinates of U in Q)
you can see that the point (-6,2) is mapped onto (-6,4) and (0,2) to (0,4)
soo the equation is
(-6 0) * (a b) = (-6 0)
(2 2) (c d) (4 4)
if you take (-6 0)
(2 2) to other side then it becomes the inverse...
so you multiply
(-6 0) (-6 0)
(4 4) with the inverse of (2 2)
therefore u get the answer (1 0)
(0 2)
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first write down 2 sets of co-ordinates from triangle T in a matrix form
then multiply it with the unknown matrix and equate it to the matrix form of two corrseponding sets of co-ordinates of the image i.e R
(2 8 ) * (w y) (-4 -4)
(4 4) (x z) = (-2 -8)
therefore you can find the values of w,x,y and z by multiplying (-4 -4) with the inverse of (2 8 )
(-2 -8) (4 4)
=> so the answer is (0 -1)
(-1 0)
plz infrm me if u didn't understand...as i am available anytime in studentforums...
Good luck
thanks eddie i got it rep for u.
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hey thanks a lot shan....
@priceless and reishamix
i wud like to learn the base vectors method ..plzz.
it wud very kind of u to teach me..
thank you
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can someone plzz teach me base vectors..!!
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draw like a sketch of an axis
mark a point "i" wid de coordinates (1,0)
and "j" wid de coordinates (0,1)
thts the base vector...
theyr written like (1 0)
(0 1) like any coordinates
and whn they ask u to change the object into the transformation (0 1)
(1 0)
u simply change the coordinates on "i" and "j"
and u knw if its a reflection , rotation or wtevr!!
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theyr easier to use
and very simple
i cant draw em here gimme ur email adress
and ill draw em on msn
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reishamix by any chnce do u study in manarat? cuz thts xactly how our teacher taught us
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here is my solution (the previous one had a mistake)!
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lol no way
i dnt evn live in a city
i go to britich intnl school in tabuk
which is more like a village
and we have everything here
xcept a school building , book n teachers
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haha nice one
a pdf for that
that made me laugh
u made it? [then it is very good]
but a 10kb pdf made me laugh
nice one
+rep
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please send it 2 me on my email zahraamohd_94@hotmail.com ;)!!!
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hey thanks a lot shan....
@priceless and reishamix
i wud like to learn the base vectors method ..plzz.
it wud very kind of u to teach me..
thank you
Lol ya sure np. umm ok here v go.
Reflection:
1. reflection in ml y=x it's (0 1)
(1 0)
2. refelction in ml y=-x itz (0 -1)
(-1 0)
3. refelction in x-axis itz (1 0)
(0 -1)
4. reflection in y-axis itz (-1 0)
(0 1)
Rotation
1. rotation +90 degrees(anticlockwise) around centre(0,0) itz (0 -1)
(1 0)
2. rotation -90 degrees(clockwise) around centre (0,0) itz (0 1)
(-1 0)
3. rotation 180 degrees around centre (0,0) itz (-1 0)
(0 -1)
Enlargement
1. centre (0,0) wer k=scale factor, itz (k 0)
(0 k)
Shear
1. x invariant with x-axis wer k=scale factor (1 k)
(0 1)
2. y invariant with y-axis wer k=scale factor (1 0)
(k 1)
Stretch
1. x invariant with y-axis wer k=scale factor (1 0)
(0 k)
2. y invariant with x-axis wer k=scale factor (k 0)
(0 1)
phew don't LOL i no itz a lot but once u no dem itz rely easy. u can du ur sums very easily den widout doin da point co ordinate thng. once u no wat da transformation is eddie_adi619 u ll no wat da matrix is also. gud luck 2 all 4 ur xams hope v all du well.
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i hope u understand dem n hope i helped. gud luck evry1!!! ;)
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NOO!
I was just typing those things when you posted it!
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LOL sory. can u just explain me da shear n stretch base vectors. i no dem but dont get dem....pls help sweetsh.
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have to say lol ;D
when it is parallel to x axis, then y axis is parallel
and vice versa
but then howcome
how come in shear
when it is parallel to x axis, x axis is invariant. how :S??????
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LOL sory. can u just explain me da shear n stretch base vectors. i no dem but dont get dem....pls help sweetsh.
what is base vectors, please help!
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have to say lol ;D
when it is parallel to x axis, then y axis is parallel
and vice versa
but then howcome
how come in shear
when it is parallel to x axis, x axis is invariant. how :S??????
im sory but wat du u mean wen itz parallel 2 x-axis den y axis is parallel. hw can both axes b parallel? lol im sooo confused n bout da base vectors dey r da matrixes i wrote down-dey r called base vectors cuz dey r da same evry tym but shear n stretch just duznt go in2 my head AAAAA LOL ??? :-\
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leave it!
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LOL is dat posible but i dunno wat 2 du. i guess i ll try 2 sk my teachers or sumthng. neway can i sk u a ques? ummm r u nline or not cuz u r replying but ur status says ur offline..?LOL
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hey..the ones whu dint quite understand the inverse methos to find the transformation matrix...it can also be don't with the help of simultaneous equations..juz multiply as if the invariables wer numbers wich wil giv yu the equations thn solve..:D
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hmmmm nvr tried dat...is it posible 2 giv an eg?
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P Q * (-3 1 -3) = (-3 1 -3)
R S (2 1 -1) (-5 0 -2)
so wen u multiply it u gt...-3p + 2q p+q -3p-q
-3r+2s r+s -3r-s
nw juzz take for eg...-3p+2q=(-3)(here im takin the number from the transformed matrix to equal it..hope ur understandin cuz i suk at explainn..)
and take the other equation p+q=1
nw solve the two as simultaneuos equations..(hope u no hwda do dat!:P juz kiddin..)
so yu gt the two values and den do the sme for the variables 'r' and 's'...there yu go:D
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nice method..
thank u
+rep for u :)
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my pleasure.. ;D
all the best to all yu guys out there..hopin a*s for all.. ;)
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P Q * (-3 1 -3) = (-3 1 -3)
R S (2 1 -1) (-5 0 -2)
so wen u multiply it u gt...-3p + 2q p+q -3p-q
-3r+2s r+s -3r-s
nw juzz take for eg...-3p+2q=(-3)(here im takin the number from the transformed matrix to equal it..hope ur understandin cuz i suk at explainn..)
and take the other equation p+q=1
nw solve the two as simultaneuos equations..(hope u no hwda do dat!:P juz kiddin..)
so yu gt the two values and den do the sme for the variables 'r' and 's'...there yu go:D
lol thnkz... our teacher taut us this but nvr used it, hmmm i guess it wud b helpful. thnkz newayz cuz i just understud it b8r n actualy ur a pretty gud explainer lol, not u but i suck @ xplaining LOL :D n yes hopefully v all wud get A*s 4 sure!!!!! Best of luck evry1!!! ;D
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yeah..n e tym.. 8)
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hey thanks a lot shan....
@priceless and reishamix
i wud like to learn the base vectors method ..plzz.
it wud very kind of u to teach me..
thank you
Lol ya sure np. umm ok here v go.
Reflection:
1. reflection in ml y=x it's (0 1)
(1 0)
2. refelction in ml y=-x itz (0 -1)
(-1 0)
3. refelction in x-axis itz (1 0)
(0 -1)
4. reflection in y-axis itz (-1 0)
(0 1)
Rotation
1. rotation +90 degrees(anticlockwise) around centre(0,0) itz (0 -1)
(1 0)
2. rotation -90 degrees(clockwise) around centre (0,0) itz (0 1)
(-1 0)
3. rotation 180 degrees around centre (0,0) itz (-1 0)
(0 -1)
Enlargement
1. centre (0,0) wer k=scale factor, itz (k 0)
(0 k)
Shear
1. x invariant with x-axis wer k=scale factor (1 k)
(0 1)
2. y invariant with y-axis wer k=scale factor (1 0)
(k 1)
Stretch
1. x invariant with y-axis wer k=scale factor (1 0)
(0 k)
2. y invariant with x-axis wer k=scale factor (k 0)
(0 1)
phew don't LOL i no itz a lot but once u no dem itz rely easy. u can du ur sums very easily den widout doin da point co ordinate thng. once u no wat da transformation is eddie_adi619 u ll no wat da matrix is also. gud luck 2 all 4 ur xams hope v all du well.
Can you please specify which transformations are multiplied and which are added?
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lol sweetsh u dont hv 2 multiply any matrixes or add any. just know wat da transformation is n write down da base vector. dats it. gud luck!!!
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LOL! I dont like this word at all but i'll use it in this situation.
You didnt inderstand me my friend, sometimes they give you points and then you have to draw them under this transformation, so you have to multiply for example for rotation (0 -1)
(1 0)
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matrix *object=image
v multiply n nt add....
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hope u got wat i meant to explain....
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Oh gosh im stupid in this topic! Im an A* in Math but the transformation thing!
Yes I know that zara sweety, but for enlargement you add.
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ok i wudnt use da word if u dont like it but ur ryt i didnt understnd u @ first but now i du. ummm ur asking wen u hv 2 find da new coordinates ryt?..... hmmm i wud hv told u but i dnt use dat method cuz i usually just no wat da transformation is 4rm da base vector given n den use a tracing paper 2 du it as v r allowed 2 use tracing papers itz much more easier n saves tym also. ;)
so i suggest u 2 sk sum1 els cuz if i new i wud hv definitely xplained it 2 u...gud luck neway!!! :)
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so u no wer 2 add n multiply so wats da prob sweetsh? ???
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I really dont know!
I was confused things are getting better. THANK YOU!
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hmmm i no happens cuz evn im an A* math student but sumtymes may God help us.... ;) n ur most welcum though i shud b da 1 telng u a very big thnku as u hv helped me alot. THANKZ n all da best!!!!
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@sweetsh
only in translation u add
M=Matrix
I= iamge coordinate
O=object coordinate
for translation only
I=M+O
for all other
enlargement,rotation reflection..
it is
I=M.O
note the M is the 2x2 matrix and O coordinater(lets take 3 coordinate(a trinalge coordiante) for example
so then it will be (rotation 90 clockiwise that is)
I=(0 1) X (1 2 3)
(-1 0) (4 5 6)
and yeah no need to learn all the thing just draw this and remember
I
I(0,1)
(-1,0)------------(1,0)
I(0,-1)
I
if it is 90 clockiwise then 0,1 will be 1,0
and 1,0 will be 0,-1
NOTE:- I is not image coordinate..
I is the the straight line of y axis
and ---- is the x axis line
+ rep if u find it helpful:D