first problem.
a. prove the four vectors in W_1 and W_2 span a space of dimension 3.
b. u have redundant vector, so representation is not unique
c. just do formula P = W (W^TW)^{-1}W^T
d. easy
e. any vector in W_1 plus another vector in W_2

second problem is easy. u only need to prove:

a. if w_1 and w_2 are in W, then w_1 +w_2 is also in W. trivial from defiintion of eigenvalue
b. if w_1 is in W and c is any scalr, then c w_1 is also in W. also trivial.

非常谢谢你的帮助。
第一题的 e， 我觉得有点问题，我觉得 a(1,0,1) 这种的vector 满足w=p1w +p2w, (1,0,1) in W2
2# Enthuse

3# fromfan

you are right. i was too quick, my answer to 1e is incorrect.

you need to solve the equations. The answer should not be unique.

4# Enthuse
还是很谢谢啦，过会我会把完整答案附上

答案如下：