Representing transformations by matrices

Transcription

1 Teachig Further Mathematics FP Give each pair of studets a copy of the sheet below elarged oto A. Represetig trasformatios by matrices Studets have to multiply the matri by the positio vector of each verte of the triagle, plot the ew positio of the triagle ad state the trasformatio that the matri represets. This eables studets to fid out which trasformatio each matri represets. Discussio poits: Compare y y Which trasformatio is this? What happe to the poit with positio vector?...?...? What happe to the poit with positio vector? Is this always the case? is a reflectio i the lie. This could be epressed y y Which trasformatio is. What is the correspodig matri? y y How would you write the rotatio of 8 trasformatio i the form? What is the correspodig matri trasformatio? y Susa Wall, MEI 6 Page of

2 Teachig Further Mathematics FP Susa Wall, MEI 6 Page of (, ) (, )

3 More Matri Trasformatios Fill i the matrices for the followig trasformatios A B C D E F G H I J K L M Trasformatio Matri Trasformatio Matri Reflectio i the N Rotatio by 45 clockwise about Rotatio by O Reflectio i the y clockwise about Stretch with scale P Elargemet with factor 5 parallel to y scale factor Reflectio i the lie Q Rotatio by 8 y Rotatio by R Reflectio i the lie 45 aticlockwise y Rotatio by S Rotatio by 9 aticlockwise clockwise Rotatio by T Rotatio by 5 clockwise about aticlockwise Reflectio i the lie y U Rotatio by 5 clockwise Rotatio by V Rotatio by 6 9 clockwise about Stretch with a scale W Reflectio i the lie factor 4 parallel to y Reflectio i the lie X Rotatio by y 5 aticlockwise Rotatio by Y Elargemet with a 6 clockwise about scale factor Rotatio by Z Reflectio i the lie aticlockwise y Fid the smallest positive value of i each of the followig cases:. 6. L I. H I 7. M X I. I 8. V Q I 4. I 9. A O I 5. I. G T I I

4 Solutios to More Matri Trasformatios Trasformatio Matri Trasformatio Matri A Reflectio i the N Rotatio by 45 clockwise about B Rotatio by O Reflectio i the y clockwise about C Stretch with scale factor P Elargemet with scale 5 parallel to y 5 factor D Reflectio i the lie Q Rotatio by 8 y E Rotatio by R Reflectio i the lie 45 aticlockwise y F Rotatio by S Rotatio by 9 aticlockwise clockwise about G Rotatio by T Rotatio by 5 clockwise about aticlockwise H Reflectio i the lie U Rotatio by y 5 clockwise about I Rotatio by V Rotatio by 9 clockwise 6 J Stretch with a scale 4 W Reflectio i the lie factor 4 parallel to y K Reflectio i the lie X Rotatio by y 5 aticlockwise L Rotatio by Y Elargemet with a 6 clockwise about scale factor M Rotatio by Z Reflectio i the lie aticlockwise y

5 Teachig Further Mathematics FP Geometrical properties of matri trasformatios: y y y y y y + y y y y y y y y y y y + y + y y + y y y y + y y y + y Projects a poit dow oto the - Projects a poit across oto the y- Projects a poit up/dow oto the lie y Leaves the poit uchaged Reflects i the lie y See eample below + y Notice how the lie y collapses to the y + y origi Ivestigate etries of, ad A rotatio through 8º. y y MEI 6 Page of

6 Teachig Further Mathematics FP Matrices Usig the matrices give o the sheet below, ask studets which they ca combie ad i what order, to make the followig trasformatios: Rotatio of 8º Idetity Reflectio i the - Reflectio i the lie y Ask them if they ca make ay of them i more tha oe way. They should justify their aswers usig matri multiplicatio. Discussio poits: What is the trasformatio correspodig to? How do you kow? How about?...ad? ad? Eample: Method. Method. y y y y y y I Method above, how does this show that the trasformatio is a reflectio i the lie y? What is the relevace of the itermediate step i Method? What are the idividual trasformatios ad? I terms of what these idividually do, covice me what their combied effect is. How else might you covice yourself of the ature of the resultig trasformatio. (You might suggest they thik what happes to ad uder the combied trasformatio.) Susa Wall, MEI 6 Page of

7 Teachig Further Mathematics FP Susa Wall, MEI 6 Page of

8 MEI FP, Jue 9 9 You are give that M, ad N Q. (i) The matri products Q(MN) ad (QM)N are idetical. What property of matri multiplicatio does this illustrate? Fid QMN M, N ad Q represet the trasformatios M, N ad Q respectively. (ii) Describe the trasformatios M, N ad Q A C O - B 4 5 Fig. 9 (iii) The poits A, B ad C i the triagle i Fig. 9 are mapped to the poits A, B ad C respectively by the composite trasformatio N followed by M followed by Q. Draw a diagram showig the image of the triagle after this composite trasformatio, labellig the image of each poit clearly. 9(i) 9(ii) Matri multiplicatio is associative MN MN QMN M is a stretch, factor i the directio, factor i the y directio. B M A A(ft) B B Attempt to fid MN or QM or QM Stretch factor i the directio Stretch factor i the y directio N is a reflectio i the lie y. B 9(iii) Q is a rotatio through B M A Applyig matri to poits. Mius each error to a miimum of. B Correct, labelled image poits, mius each error to a miimum of. Give B4 for correct diagram with o workigs.

9 MEI FP, Ja 6 9 A trasformatio T acts o all poits i the plae. The image of the geeral poit P is deoted by P. Every poit P is mapped oto a poit P o the lie y. The lie joiig a poit P to its image P is parallel to the -. (i) Write dow the image of the poit (,5 ) uder trasformatio T. [] (ii) P has coordiates (, y ). State the coordiates of P. [] (iii) All poits o a particular lie, l, are mapped oto the poit (, 6 ). Write dow the equatio of the lie l. [] (vi) I part (iii), the whole of the lie l was mapped by T oto a sigle poit. There are a ifiite umber of lies which have this property uder T. Describe these lies. [] (vii) For a differet set of lies, the trasformatio T has the same effect as traslatio parallel to the -. Describe this set of lies. [] (vi) Fid the matri which represets the trasformatio. [] (vii) Show that this matri is sigular. Iterpret this result i terms of the trasformatio. [] 9(i) 9(ii) ( 5,5 ), yy B [] B, B [] 9(iii) y 6 B [] 9(iv) 9(v) 9(vi) 9(vii) All such lies are parallel to the -. All such lies are parallel to y. det B [] B [] B [] M Mius each error Trasformatio may to oe. E [] Give mark for partial eplaatio

10 MEI FP, Ja 7 9 Matrices M ad N are give by M ad (i) Fid M ad N. N. 4 (ii) Fid MN ad ( MN ). Verify that (iii) The result MN N M. [6] PQ Q P is true for ay two square, o-sigular matrices P ad Q. The first two lies of a proof of this geeral result are give below. Begiig with these two lies, complete the geeral proof. [] PQ PQ I PQ PQQ IQ 9(i) M N 4 7 M A A [] Dividig by determiat Oe for each iverse 9(ii) 5 MN 4 4 M A 4 MN 5 A N M ( MN) M A A [6] Multiplicatio Statemet of equivalece to ( MN ) 9(iii) ( PQ) PQ PQQ IQ PQ PI Q PQ P Q PQ PP Q P PQ I Q P Q P M A M A QQ I Post-multiply by P

11 MEI FP, Ja You are give that A k ad B 5 + k ad that AB is of the form α 4k 8 AB 5k 6 + 9k + 6 k. β (i) Show thatα ad β k. [] (ii) Fid AB whe k. (iii) For the case whe k write dow the matri [] - A. [] (iv) Use the result from part (iii) to solve the followig simultaeous equatios. + 4y z y+ z 9 + 7y z 6 (i) α ( k) β k B M A [] (ii) 4 AB 4 4 B [] Mius each error to mi of (iii) A M B A Use of B 4 Correct iverse [] (iv) y z , y, z M B A Attempt to pre-multiply by s.c. B or B (see performace) for Gaussia elimiatio Mius each error