5). Let V be the space of meromorphic functions on the extended complex plane Ĉ that possibly have poles at 0, 1 and ∞ with orders up to 2. Find a basis for V.

(d) Passing through the point (0, 1, 2), perpendicular to the line L and intersects L. L(xo, v) = TER

1. Let V be the set of all ordered pairs of real numbers, and consider the following addition and multiplication operations on -() and r-(.2): +Y (M +V+1, +v +1). ki - (ku ku) Is Va vector space? Justify your answer

. Show that if v is orthogonal to both w1 and w2, then v is orthogonal to k1w1 + k2w2 for all scalars k1 and k2.Also, Use graphical representation to show how v is orthogonal to the plane formed by  w1 and w2

Let S be the space of all complex vectors (x, y, z) such that y = z². Determine whether or not S is a linear space.

10. Let {u1, · .. , Un} be a list of vectors in an n-dimensional vector space V and let B be a basis for V. Let S = {[u1]B,· , un]B} be the list of coordinate vectors of {u1,, Un} with respect to basis B. Prove that span{u1, .. , un} = V if and only if span(S) = R".

Can any of the given sets of 3-vectors below span the 3-space? Why or why not? (a) [121] [231] [342]

a) Any vector z ER" can be writ ten as a line ar combination of the stand ard basis vectors e1, e2; *. en > akek k=1 Determine the coefficients ar in terms of z and eg only.

1. Let V be R2, the set of all ordered pairs (x, y) of real numbers. Define vector addition and scalar multiplication in V as (x1, y1) + (x2, Y2) = (x1 + x2, 0) and r(x1, y1) = (rx1,ry,) Is V a vector space? If not, list all the properties which fail to hold.

1. Let V₁, V2,...,Vn be n vectors in a vector space V. Explain what it means to say that these vectors span V.