Problem 66. Give an example of a linear map T : R2 + R2, such that dim KerT = 1. Give an example of a linear map T : R' + R', such that dim KerT = 1. Give an example of a linear map T : R* + R', such that dim ImT = 1. Is there a linear map T : R' + R', such that dim Im T = 1 and dim KerT = 1? Read chapter III in the text with understanding. Show that the functions fi(x) = 1 + x and f2(x) = 1' are linearly independent. Show that the functions fi(x) = el and fa(x) = 1 + 23 are linearly independent. Show that the functions fi(x) = et and fa(x) =0 are not linearly independent. Give an example of two non-zero functions fi(x) and fe(x) that are not linearly independent. Read Example 4 on page 7 and Example 2 on page 10 in the text with understanding