1. Find ii-B and the angle between and ii for ii- -2 and 1 2. Determine whether

Question

1. Find ii-B and the angle between and ii for ii- -2 and 1 2. Determine whether S- (,y, )y0 is a subspace of V-R3 (a) Is S closed under vector addition? (b) Is S closed under scalar multiplication? (c) Is S a subspace ofV? Explain. 6 3. Consider the vectors = and Tig o in R3. (a) Are the vectors linearly independent? Explain. (b) Do they form a basis for R? Explain Find a basis for the nullspace of A 2 2 5. Find a basis for the vector space of all vectors in R whose components add to zero. What is the dimension of this vector space? 6. Find the projection of u- 7. Let W be the plane in R3 that contains the line y in the ry-plane and also contains the z-axis. Find a basis for W. What is the dimension of W? Can you extend your basis to a basis for all of R Explain.

Explanation / Answer

3.

a)

No. Because , u2=-3u1

b)

No. R3 has dimension 3 so a spanning set must have at least 3 vectors but there we have only 2 vectors u1,u2 hence then do not span R3 and hence do not form a basis

4.

Let, x=[a b c]^T be in null space of A

SO, Ax=0

This gives us three equations

a+b-c=0

a+2b+2c=0

2a+3b+c=0

Note that adding first two equations gives us the third equation

So we have two independent equations and 3 variables. So there is one free variable

We choose c to be the free variable

So,

a+b=c

a+2b=-2c

Solving for a,b in terms of c gives

b=-3c, a=4c

x=[4c -3c c]^T

So basis for nullspace of A is

{[4 -3 1]^T}

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