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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.

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