For each item below decide if the given subset W of V forms

For each item below, decide if the given subset W of V forms a subspace of V. If it is a subspace, find a basis for W. Otherwise, explain why W does not form a subspace of V. a. V = P_4(R); W = span{1 - 3x^2 + 2x^4, x^2 - 3x^4, 2 + x^4, 1 + x^2 + x^4} b. V = 2 times 3; and = {[a b c d e f]: a + b = c + f and a - c = e - f - d}

Solution

(a).We have V = P₄(R) and W=span{1-3x²+2x⁴, x²-3x⁴, 2+x⁴,1+x²+x⁴}. Let X= a(1-3x²+2x⁴)+b(x²-3x⁴ )+ c(2+x⁴)+ d(1+x²+x⁴) and Y = e(1-3x²+2x⁴)+f(x²-3x⁴ )+ g(2+x⁴)+h(1+x²+x⁴) be 2 arbitrary vectors in W, where, a,b,c,d,e,f,g,h are arbitrary real numbers and let k be an arbitrary scalar. Then X +Y = a(1-3x²+2x⁴)+b(x²-3x⁴ )+ c(2+x⁴) + d(1+x²+x⁴) + e(1-3x²+2x⁴)+f(x²-3x⁴ )+ g(2+x⁴)+h(1+x²+x⁴) = (a+e) (1-3x²+2x⁴)+ (b+f) (1-3x²+2x⁴)+(c+g) (2+x⁴)+(d+f) (1+x²+x⁴). Thus X+Y W. Further kX = k[a(1-3x²+2x⁴)+b(x²-3x⁴ )+ c(2+x⁴)+ d(1+x²+x⁴)] = ka(1-3x²+2x⁴)+kb(x²-3x⁴ )+ kc(2+x⁴)+kd(1+x²+x⁴) so that kX W. Also, W contains the zero vector. Hence W is a vector space, and therefore, a subspace of V.

(b). What is to be done here has not been stated. Please upload again.