I I. veil. Determinewhathappenstothestandardbasisvect. tea. find their matrices The name of the game. Stretches. What is L

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1 3.3 Geometr.ca Transormati Many linear transormations in R2 and R are easy to visualize and play an important role in modelling and structural analysis Here we record several such transormations and ind their matrices The name o the game Determinewhathappenstothestandardbasisvect Stretches Let s t be 70 and consider the linear transormation 2 that stretches vectors Ee R by S in the n direction and t in the Nz direction What is L Well 451 set so veil tea o

2 Exe Lik R is a stretch by 2 in the Xi direction and a contraction by in the xz direction then D Zo UZ a e a t l L UZ a L stretches contracts the Xi and Xa directions by the same actor t so L call L a stretch too t we contractionbyaactorot n 1123 i L stretches or contracts the x X and Xs directions by r s t µtt respectively simply then

3 nepart.eu pe a stretch by a actor o 1 leaves every rector unchanged This transormation id R R is idcx given by and or this reason we call it the identity Since id ei e i its matrix is id E et En n Projections perpendicularse n 1.4 we saw that i t 8 then or all Proj t E Prog t Proj E Proj tj tproj y g c R and all te R Thus the map Pooja R R that sends yo to Proj is linear 1

4 ts matrix Proje can be ound by calculating Project Project Projy En see Q2 o Assignment 8 see ma Proj a e n T k Rotations a se Let Ro R 1122 be the transormation that rotates all vectors counterclockwise about the origin by 0 radians Note t doesn't matter i a vector is stretched and then rotated or rotated and then stretched i e Ro tx trocx so Ro satisies L2

5 Furthermore i and j tg rotates with K and j n se are vectors then as we apply Ro Ro ath i e so it doesn't matter i we irst add and then rotate or rotate and then add i.e Rolxtyt Rohittrotyl Ll holds so Ro.is inear What is Ro Answering this question is easy or some angles Ra Ke 12 see a E a l i Fr 4 r i i r se

6 Rahleil 19 RCE Raleil o Raceil What about or a From that our Ra t general 0 sea t Raa it lra o Y knowledge o the unit circle we know Rohit 5 8

7 What's Ro ej t should be the same as R LeT cos Ot Note costatp cosacosp Sind sinp Sin Ot Coso cos Tla Sin sixty siroccos th t cos sin Th L t c sink B sins cos p Cosas np 3 Thus Cro Roca roles L Ex Determine Ray and use it to ind the vector obtained by rotating E 2 radians counterclockwise 4 by Z

8 We have solution r cos sin R H sin i ed Raz Y ii Relectionoveralineinpelet's say through the cos we have a line in 1122 that passes origin vector or the line so A o and suppose 5 is a normal direction vector

9 o Rel R R2 denotes the map that relects each vector over this line then th g 1Rel 2Proj txt Y projn i L Rel L K since Reli K 2Projn.C lid ZProjn lx Since id and Proj are linear so is this we have that Reli R R is linear and Rel id 2Proji 21Proja E Find the matrix that represents a relection across the Xr axis in R2 Solution We can see right away that this map send e to and a to g

10 Let's see i our above approach gives the same result ANZ e a so we get t Since We n have t T plz lo is 1 to this line we're looking or Project 1, and This means Praja Rel 2Proj tog Projekt o g and hence Ret 2Proja 21 Yay it t's the same

11 E Find the matrix that represents a relection Solution We have across the line t te R Use this matrix to ind the vector obtained by relecting n 2 in this line is a normal vector Projn E Projiei E So Projn Yg and hence Reth 21Proji 2 Thus Reli ,1 19 7

12 Relectionoveraplanein Let's say we have a plane in R that passes through the origin and has normal vector rt now Reli R R denotes the map that relects vector every 5 over this plane in A then Rel 2Proj txt This means once again Re i2oj projnxxf.ee i4r i Y Rel 9 J E Consider the plane X 2x X 0 in RS Find the matrix representing a relection in this plane then ind the image o 13g under this map

13 Solution The normal rector is 5 14 and we must ind Reine 2 Pooja since Projet L Projiei Projiei tl tp we have Proji and hence Rel 2 Proji t it ih The relection o 13g in this plane is retiedd Kettell o t.tt