Question

Problem 13. Let l be the line in R' spanned by the vector u = 3 and let P:R -R be the projection onto line l. We have seen that projection onto a line is a linear transformation (also see page 218 example 3.59). a). Find the standard matrix representation of P by finding the images of the standard basis vectors e, e, and e, under the transformation P. b). Find the standard matrix representation of P by the second method (using matrix multi- plication and transpose) as well. c). Find the kernel and range of P. d). Let Q:R - R'be defined as Q=P2-P. It can be easily shown that is also a linear transformation (for the sake of practice you should try to verify this). Find the kernel and range of Q. Are you surprised by the answers you have got? e). Let R be the transformation that reflects vectors across the line 1. Let XER', then its reflection across the line l is the vector R(x) (image of x under R) and its projection onto line l is P(x). What is the geometrical link between the vectors x, R(x) and P(x)? f). Using the geometric link you obtained (in part (e)), find the images of the standard basis vectors e, e, and e, under the transformation R. g). Find the standard matrix of the reflection transformation R.

Answer 1

(4/6.2) 14 2,3,) 7 Puoja Pu) e 14 (2,3,) 14 4 69 3 3 2 Matwr la7 462 693 2 3 4 62 693 2 3 P- a(aTu)4T 14 Spar Rnge ( Keurpa 24-22 Kange O