Problem 20. In this question, we'll consider linear transformations in R2 and R3 that transform familiar geometric shapes. 12 (a) Find an invertible linear transformation that maps the unit circle x² + y² = 1 to the ellipse 1½ x² + 1½³ y² = 1. By "maps the circle to the ellipse", we mean that every vector lying on the circle is sent to a vector lying on the ellipse. Is your answer unique? Justify rigorously! (b) Find a linear transformation that maps the unit sphere x² + y²+z² = 1 to the ellipsoid x²+ y²+ z2 16 = 1.

The right triangle in the coordinate plane is rotated 270° clockwise about the point (2, 1) and then reflected across the y-axis to form triangle A'B'C". The pre-image of triangle ABC has coordinates A(2,1), B(2,4), and C(6,1). After performing these transformations on graph paper (using tracing paper if needed), state the coordinates of the image of point A after the transformations. A' = (

(2) Can the change of cartesian coordinates to polar coordinates in R? be expressed as a linear transformation? Why or why not?

Find the reflection of the point (1, 2, 3) in the point 2x+y=10