Question 27384
Let V = P3(Q) and W = { f(x) 2 V | f(x) = f(−x + 1) }. Find a basis for W. 
Let V and W be same as above. Find a basis for V that contains a basis for W. 
[Usually, P3(Q) means the vector space of polynomials of deg <= 3 in Q[x])

Sol: If f(x) ={{{ax^3+bx^2+cx + d}}} belongs to W , then f(x) = f(-x+1) implies {{{ax^3+bx^2+cx + d = a(-x+1)^3+b(-x+1)^2+c(-x+1) + d }}} or {{{ax^3+bx^2+cx + d }}}= {{{a(-x^3+ 3x^2 - 3x -1)+b(x^2-2x+1)+ c(-x+1) + d}}}= {{{-ax^3+ (3a+b)x^2+(-3a-2b-c)x - a + b + c + d }}} 
    By comparing the coefficients, we have a=-a,  3a+b = b, -3a-2b-c=c ,    d= –a+b+c+d. Hence, a= 0, b+c = 0. We see that W = { {{{b(x^2- x) + d }}} | b,d in Q} (generated by {{{x^2-x}}} & 1and so dim W = 2. By choosing B’ = {1, {{{x^2-x}}} } we get a basis of W. The adjoining two independent vectors {{{x^3 }}}and x (not in W) to B', then we can get a basis B= {1, {{{x^2-x}}}, {{{x^3}}}, x} of V, which contains a basis B’ for W.

 Kenny