I need to find a basis of V that includes u and v I need to

I need to find a basis of V that includes u and v.

I need to find a basis of V that includes u and v. U = x^2 + 1 and u = x^2 + x in V = P_3(R^3). u = 1 - x^2 and u = 1 + x^2 in V = P_2(R).

Solution

V = P3( R3) is the vector space of polynomials upto degree 3. An arbitrary element of V is of the form ax3 +bx2 +cx +d., where a, b, c, d are arbitrary scalars Let w = x3 and z = 1. Then the vectors u, v, w, z are linearly independent as a linear combination of these vectors cannot be 0 without all the scalar coefficients being 0. Further, since ax3 +bx2 +cx +d. = a(x3) + (b-c)(x2 +1) + c(x2 +x) + (d- b+ c)1 = aw+ (b-c)u + cv+ (d-b+c)z. Hence { u, v, w, z } forms a basis for V. Here, V = P2( R3) is the vector space of polynomials upto degree 2. An arbitrary element of V is of the form ax2 +bx +c. Let w = x. Then u, v, w are linearly independent as a linear combination of these vectors cannot be 0 without all the scalar coefficients being 0. Further , ax2 +bx +c = [(a+c)/2](1+x2) + bx + [(c-a)/2](1-x2) = [(a+c)/2]v +bw + [(c-a)/2] u. Hence { u, v, w} forms a basis for V.