Linear Algebra question Let V = { p(x) E P2 : p(0) = -p’(1) } a) find a basis fo

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Linear Algebra question
Let V = { p(x) E P2 : p(0) = -p’(1) }
a) find a basis for V
b) for what values of k is p(x) = 6x + k in V
c) For the polynomial p(x) in part (b), is p’(x) in V? Is p”(x) in V? Linear Algebra question
Let V = { p(x) E P2 : p(0) = -p’(1) }
a) find a basis for V
b) for what values of k is p(x) = 6x + k in V
c) For the polynomial p(x) in part (b), is p’(x) in V? Is p”(x) in V? Linear Algebra question

a) find a basis for V
b) for what values of k is p(x) = 6x + k in V
c) For the polynomial p(x) in part (b), is p’(x) in V? Is p”(x) in V?

Explanation / Answer

Let p(x) = a+bx +cx2 . Then p’(x) = b+2cx so that p(0) = a and p’(1) = b+2c.

(a). If p(0) = -p’(1) , then a = -(b-2c)= -b+2c so that p(x) = -b+2c + bx +cx2 = b(-1+x) +c(2+x2). Thus, a basis for V is { -1+x, 2+x2}.

(b). If p(x) = 6x + k, then c = 0,b =6 and k = -b+2c = -6. Thus, k = -6.

(c). For the polynomial p(x) in part (b) i.e. when p(x) = 6x + k = p(x) = 6x -6, p’(x) = 6 and p’’(x) = 0. Apparently, the 0 polynomial is in every vector space so that p”(x) is in V. Further, the vector (6,0,0)T cannot be expressed as a linear combination of the vectors (-1,1,0)T and (2,0,1)T so that p’(x), as in part(b) (i.e.6) is not in V.

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