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Question
These problems are giving me lots of problems. They are the remaining ones I have left for the review for my exam this week.

1)          Determine if W is a subspace of R^3 and check all correct answers below.

A. W is a not vector space because it is not closed under the addition.
B. W is not a vector space because it is not closed under scalar multiplication.
C. W is not a vector space because it does not have a zero element.
D. W is a vector space because it is the solution set of a homogeneous linear system.

For this problem, i have tried B and C as seperate answers but the teacher says that they are wrong. I have tried to use letter C because W is not a vector space because it does not have a zero element. Also, i have tried using letter B because W is not a vector space because it does not have a zero element.
What would you suggest as possible answers? There is a possibility of the answer being multiple letters.

2)          Find a basis of the subspace of R^5 containing all that satisfy 5x_1+9x_2-40x_3=102x_4-76x_5=9x_2+30x_4-40x_5

How would I go about finding the 3 vectors? I know that i need to make a matrix and then solve the matrix in order to determine the basis( from the free variables after gaussian elimination) What values do i have to use in order to get a possible answer?

3)I am really clueless about what to do for this problem. I have no idea where to start.

Suppose vector 1, vector 2, vector 3 is a set of vectors mutually perpendicular. Assume that

Length of vector v1 = sqrt(35)

length of vector 2 = sqrt(94)

length vector v3 = sqrt(36)

Let w be a vector in Span(v1,v2,v3) such that

(vector w)dot product(vector v1) = 35

(vector w)dot product(vector v2) = -188

(vector w)dot product(vector v3) = 36

Then w = _____v_1+____v_2+_____v_3

1) I can't answer this because I don't know how W is defined.

2) Let x_1 = a, x_2 = b, x_3 = c.
This gives us 5a + 9b - 40c = 102(x_4) - 76(x_5) and
5a + 9b - 40c = 9b + 40(x_4) - 40(x_5).

These equations can be rewritten as
102(x_4) - 76(x_5) = 5a + 9b - 40c and
40(x_4) - 40(x_5) = 5a - 40c.

The 2nd equation can be solved for x_5, and that put back in the 1st to solve for x_4.
Putting x_4 into the 2nd equation allows x_5 to be solved for.

3) Asume v_1, v_2, and v_3 are mutually perpendicular.
Take |x| = length of vector x.
We are given |v_1| = sqrt(35), |v_2| = sqrt(94), and |v_3| = sqrt(36).
Let w be a vector in Span(v_1,v_2,v_3).

We are given w•v_1 = 35, w•v_2 = 180, and w•v_3 - 36.

Then w = _____v_1+____v_2+_____v_3

Assuming v_1, v_2, and v_3 are simple vectors, it can be found that
v_1 = <1,3,5>, v_2 = <2,3,9>, and v_3 = <2,4,4>.
We want the dot products of w with these respective vectors, in order, to be 35, 180, and 36.

That means solve
w_1  + 3(w_2) + 5(w_4) = 35,
2(w_1) + 3(w_2) + 9(w_3) = 180, and
2(w_1) + 4(w_2) + 4(w_3) = 36
for w_1, w_2, and w_3.

Volunteer

#### Scott A Wilson

##### Expertise

I can answer any question in general math, arithetic, discret math, algebra, box problems, geometry, filling a tank with water, trigonometry, pre-calculus, linear algebra, complex mathematics, probability, statistics, and most of anything else that relates to math. I can also say that I broke 5 minutes for a mile, which is over 12 mph, but is that relevant?

##### Experience

Experience in the area; I have tutored people in the above areas of mathematics for over two years in AllExperts.com. I have tutored people here and there in mathematics since before I received a BS degree back in 1984. In just two more years, I received an MS degree as well, but more on that later. I tutored at OSU in the math center for all six years I was there. Most students offering assistance were juniors, seniors, or graduate students. I was allowed to tutor as a freshman. I tutored at Mathnasium for well over a year. I worked at The Boeing Company for over 5 years. I received an MS degreee in Mathematics from Oregon State Univeristy. The classes I took were over 100 hours of upper division credits in mathematical courses such as calculus, statistics, probabilty, linear algrebra, powers, linear regression, matrices, and more. I graduated with honors in both my BS and MS degrees. Past/Present Clients: College Students at Oregon State University, various math people since college, over 7,500 people on the PC from the US and rest the world.

Publications
My master's paper was published in the OSU journal. The subject of it was Numerical Analysis used in shock waves and rarefaction fans. It dealt with discontinuities that arose over time. They were solved using the Leap Frog method. That method was used and improvements of it were shown. The improvements were by Enquist-Osher, Godunov, and Lax-Wendroff.

Education/Credentials
Master of Science at OSU with high honors in mathematics. Bachelor of Science at OSU with high honors in mathematical sciences. This degree involved mathematics, statistics, and computer science. I also took sophmore level physics and chemistry while I was attending college. On the side I took raquetball, but that's still not relevant.

Awards and Honors
I earned high honors in both my BS degree and MS degree from Oregon State. I was in near the top in most of my classes. In several classes in mathematics, I was first. In a class of over 100 students, I was always one of the first ones to complete the test. I graduated with well over 50 credits in upper division mathematics.

Past/Present Clients
My clients have been students at OSU, people who live nearby, friends with math questions, and several people every day on the PC. I would guess that you are probably going to be one more.