1. (a) Use definition of linear transformation to determine whether the transformation T: R? → R³ defined as follows is linear. T x+ 2y 2x – 4y (b) Let T : R" –→ R™ be a linear transformation. i. Is...

Extracted text: 1. (a) Use definition of linear transformation to determine whether the transformation T: R? → R³ defined as follows is linear. T x+ 2y 2x – 4y (b) Let T : R" –→ R™ be a linear transformation. i. Is it always true that T(0) = 0 where the zero vectors live in the appropriate spaces? Justify. ii. Does S(0) = 0 mean that S is linear? If not, can you give a counter-example? (c) Consider the following matrix transformation T : R² → R² defined as а b T = i. If T rotates the 2D space 150° counter-clockwise, and then reflects the result across the new r-axis, find a, b, c, d. ii. What if T reflects the 2D space across the x-axis first and then rotates 150° counter-clockwise? Do you expect to get the same a, b, c, d? Why or why not?