Let V = R' and define vector addition as (x, y, z)+(x', y', z')= (x+x',y+y',z+z') (standard addition) and scalar multiplication as k(x, y, z) = (kx,ky, 2z) (not the standard scalar multiplication). Determine if V is a vector space.

Consider the vectors u₁ = [1,0,−1], _U2 = [1,0,0] and u3 = [0, 1, 1], v = [5, 1, 3] and w = = [1,0,3]. (a) Show that the vectors u₁, u₂ and u3 are linearly independent. (b) Show that the vector v is in the span of u₁, U₂ and u3. (c) Find the coordinate vector of v with respect to {U₁, U2, U3}. (d) Write the zero vector as as a non-trivial linear combination of the vectors u₁, U2, U3 and w.

Compute the curl of each of the following vectors: V = (yz, xz, xy) F = (x, y, z) = = ( Note: Y 2 (x² + y² +2²)³/²¹ (x² + y² + x2)³/²¹ (x² + y² + x2)³/2 • If the answer is a scalar, you can just type it in the box (using the Calcpad if you like, or using / for fractions, ^ for exponents, etc.) For multiplication, you can either leave a space, or use *. So x*x=xx = x². Note that this is not the same as xx without a space; that will get read as an entirely different variable! • If you need to enter a vector, enter an ordered list of components, so for A you can enter either A = (A₂, Ay, A₂) = {Ax, Ay, Az} . Note that the system isn't great with multiplying through by overall factors, so it's better not to write e.g. • If you need to enter O, enter it in three components, as (0, 0, 0) or {0,0,0}. V x V = ▼ xr= vx= (2A, 2A, 2A₂).

2a)Use triple scalar to verify that the following vectors are not coplanar. ā=[-2,1,0], b=[-3,4,5], c =[2,0,3] b) Write a = [-4,1,7] as a linear combination of a,b,c