6. Let V = R³ and F = R. Let W = { (a, b, c) | a+b-2c=0 where a, b, c are in R} It can be shown that W is a subspace of V. (You do not have to check this.) (a) Find a basis for W and explain why it is a basis. (b) What is the dimension of W?

a b 6. Let H be a subspace of M2x2 whose vectors are of the form Then, B с 0 is a basis for H. 10 Find the coordinate vector of v = according to the basis, B.

Q7: Find a basis for the following vector spaces. a (a) V = 4b — За — d (b) W a-4e = 86+3d 2e = d

Find a basis for the plane 4x + 3y + 5z = 0 in R³. |

4 Let W = span(vV1, V2) where vi = and v2 2 . Then: (a) Find the orthogonal decomposition of the vector x = 3 (b) Find an orthogonal basis for W- (remember to justify your answer).

Let w,=(1,0,0,1) and w=(-1,0,0,1). Let W=(w | w w, = w-w, = 0}, where w-w, denotes the dot product of w, & w. (a) Show that W is a vector space (over the field R). (b) Find the basis of W. (c) Find the dimension of W. (d) Is the answer to part (c) unique?