(3+5+2+4+4+4 22 points) Let A= [1 1 1] [1 1 1]
[0 1 1]
[0 1 1]

Required:
a. Find a basis for the column space of A. What is the dimension of this basis?
b. Use Gram-Schmidt to find an orthonormal basis for the column space of A: {q1, q2, q3}
c. Find a fourth vector q4 that completes the basis you have found in part (a) to an orthonormal basis of R4. To which of the 4 fundamental spaces related to A does q4 belong to?
d. Find the projection matrix P that maps any vector onto C(A). What is the projection of b= (1, 0, 0,0) onto this space?

a.

step-by-step explanation:

step 1 distribute 2 to every term in the parentheses

step 2 add 10 on both sides of the equal sign

third choice down

step-by-step explanation:

the first inequality is based on the number of adults and campers, and the second inequality is based on the cost of feeding both the adult and the camper.

we know that there is room for 525 people max.   it's ok to have less than 525, but it's not ok to have more than 525, so the inequality is less than or equal to.   the number combination of adults (x) and campers (y) cannot exceed this 525 max.   therefore:

x + y ≤ 525

now we talk about cost.   it costs $9 to feed one adult, so depending upon the number of adults you have (x) depends upon how much it will cost to feed that number of adults.   same with the campers, who cost $5 to feed.   the adults can be represented by 9x and the campers can be represented by 5y.   since the budget cannot exceed 1,200, the second inequality is:

9x + 5y ≤ 1200