All vectors and subspaces are in r^n.

check the true statements below:

a. for each y and each subspace w, the vector y - proj_w (y) is orthogonal to w.

b. if z is orthogonal to u_1 and u_2 and if w = span{u_1, u_2}. then z must be in w.

c. the orthogonal projection y of y onto a subspace w can sometimes depend on the orthogonal basis for w used to compute y.

d. if y is in a subspace w, then the orthogonal projection of y onto w is y itself.

e. if the columns of an n times p matrix u are orthonormal, then uu^t y is the orthogonal projection of y onto the column space of u

Hye ryung is the president of the local chapter of the american medical students association (amsa). she is organizing local outreach and informational meetings. at her first meeting there are five people present (including herself). every month after that her group grows by 5 people. a. how many members are in the group after 6 months? b. the logistics of hosting and feeding her group at meetings was more complicated than expected. the cost of feeding 5 people at her first meeting was $30, for 10 people it was $35, for 15 it was $45, and after 6 months all of the costs had added up to $100. write a function to model the cost with the number of people attending meetings. c. what connections can you make between the linear growth of the group’s membership and the increasing costs of running meetings?

Pls will give brainliest and all p.s u can zoom in if u cant see plss

Quick! a survey of 57 customers was taken at a bookstore regarding the types of books purchased. the survey found that 33 customers purchased mysteries, 25 purchased science fiction, 18 purchased romance novels, 12 purchased mysteries and science fiction, 9 purchased mysteries and romance novels, 6 purchased science fiction and romance novels, and 2 purchased all three types of books. a) how many of the customers surveyed purchased only mysteries? b) how many purchased mysteries and science fiction, but not romance novels? c) how many purchased mysteries or science fiction? d) how many purchased mysteries or science fiction, but not romance novels? e) how many purchased exactly two types of books?