Answer Happy

1.9 EXERCISES In Exercises 1-10, assume that is a lincar transformation. Find the standard matrix of T. 1. TERR.7(e) = (3.1.3. 1) and (e) = (-5,2.0.0). where e = (1.0) and e; = (0.1) 2. TER-R, Tle) = (1.3). Tez) = (4.-7). and Tle) = (-5,4), where e. es, es are the columns of the 3 x 3 identity matrix 3. T: RR rotates points about the origin) through 3/2 radians (counterclockwise). 4. TER+ Rrotates points about the origin) through ---/4 radians (clockwise). (Hint: (e.) = (1/2-1/72) 3. T:R? - R is a vertical shear transformation that maps into e, -2e, but leaves the vector es unchanged. 6. T:R? - R' is a horizontal shear transformation that leaves e, unchanged and maps e into e + 3€ 7. T: RR? first rotates points through -3x/4 radian (clockwise) and then reflects points through the horizontal * -axis. (Hint: Tle) =(-1/2, 1/2). 8. T:R- R first reflects points through the horizontal - axis and then reflects points through the line 12 - *- 9. T:R? first performs a horizontal shear that trans- forms e into e: -2e Cleaving unchanged) and then to flects points through the line 12 = -1 10. T: RR first reflects points through the vertical.I-asis and then rotates points x/2 radians. 11. A linear transformation TR?-?first reflects points through the x-axis and then reflects points through the axis. Show that I can also be described as a linear transfer mation that rotates points about the origin. What is the angle of thut rotation 12. Show that the transformation in Exercise 8 is merely a rota tion about the origin. What is the angle of the rotation? 13. LTR → be the linear transformation such that (e) and Tez) are the vectors shown in the figure. Using the figure, sketch the vector (2.1). ET 41 10 doo Tle Tied म how 14. Let TR-R be a lincar transformation with standard matrix A = la: al, where a, and are shown in the figure. Using the figure, draw the image of under the