Math 33A Discussion Example Austin Christian October 23, Example 1. Consider tiling the plane by equilateral triangles, as below.

Transcription

1 Mth 33A Discussion Exmple Austin Christin October 3 6 Exmple Consider tiling the plne by equilterl tringles s below Let v nd w be the ornge nd green vectors in this figure respectively nd let {v w} be the bsis for R formed by these two vectors Let u be the purple vector () Sketch the vector in the figure (b) Give the components of u with respect to the bsis (c) Let T : R R be the liner trnsformtion tht rottes the plne counterclockwise through n ngle of π/3 Find the mtrix T 8 (d) Is the terminl point of the vector in the center of one of the red hexgons 6 or on n edge? (Solution) () ecuse this vector is given in the bsis {v w} the desired vector is equl to v + w which is the blue vector in the following figure:

2 (b) We see tht we cn get to the terminl point of u by following w nd then following v twice Alterntively we could follow v twice nd then follow w In either cse we see tht u w v (c) When we pply this rottion v nd w re crried to their dshed counterprts in the following figure: We see tht T (v) w v nd T (w) v so ( ) T nd T ( ) This mens tht the mtrix of T with respect to is given by T (d) Notice tht the vector lies in the center of hexgon: It is not difficult to see tht integer multiples of this vector will lso lie in the centers of vrious hexgons ecuse our vector is such multiple it lies in the center of the 8-th hexgon bove the hexgon contining the origin

3 Exmple ( 34 Exercise 9 of ) Let A v nd v Find the mtrix of the liner trnsformtion T ( x) A x with respect to the bsis ( v v ) in the following three wys: () Use the formul S AS (b) Use commuttive digrm (c) Construct column-by-column (Solution) () The mtrix S is the chnge-of-bsis mtrix tht we use to trnsition from the stndrd bsis to nd it hs columns v nd v So S nd S We my then compute S AS c Notice how this formul works If we hve vector in our new bsis the product c c S c v + c v tells us to consider this s liner combintion of the vectors c v nd v We then multiply by A corresponding to the ppliction of our liner trnsformtion T nd finlly multiply by S which tells us to represent liner c combintion c v + c v by the vector c (b) The commuttive digrm we hve in mind strts with n rbitrry vector x c v + c v R in the upper left corner: x c v + c v On the one hnd we cn pply the trnsformtion T to x to obtin T ( x) A x A(c v + c v ) c A v + c A v c v c v 3

4 On the other hnd we could write x s vector c c in our new bsis These two options comprise the upper right nd lower left corners of our commuttive digrm respectively T x c v + c v c v c v c c Once we ve pplied the trnsformtion T to obtin T ( x) c v c v we cn express this imge s vector in : c T ( x) c c c The mtrix T is then the mtrix tht trnsforms into c c s below: T x c v + c v c v c v c b Writing we see tht we wnt c d c c c b c c d c c So we hve b c nd d Tht is just s we found bove c c + bc cc + dc (c) Our finl pproch for finding relies on the importnt observtion tht the columns of mtrix correspond to the imges of the bsis vectors expressed in the relevnt bsis Tht is since represents the liner trnsformtion T with respect to the bsis ( v v ) the columns of re the vectors T ( v ) nd T ( v ) respectively expressed in the bsis We hve T ( v ) A v v 4

5 nd T ( v ) A v so T ( v ) These re the columns of so just s before nd T ( v ) v Exmple 3 ( 34 Exercise 7 of ) Find the mtrix of the liner trnsformtion T ( x) A x with respect to the bsis ( v v v 3 ) where 4 4 A v v nd v (Solution) The columns of will correspond to the imges T ( v ) T ( v ) nd T ( v 3 ) respectively so we compute T ( v ) A v 9 9 v + v + v T ( v ) A v v + v + v T ( v 3 ) A v 3 v + v + v Expressed in our bsis we hve 9 T ( v ) T ( v ) so the mtrix is given by 9 5 nd T ( v 3 )

6 Notice tht the usul formul should lso do the trick but the right side would be quite tedious to compute References Otto retscher Liner Algebr with Applictions Person Eduction Inc Upper Sddle River New Jersey 3 6