In each part determine whether W is a linear subspace of V J

In each part determine whether W is a linear subspace of V. Justify your answer carefully; if you think W is a subspace of V check all the necessary conditions and if you think W is not a subspace of V explain which condition fails by giving an example. V=P_2, W={W elements of P_2|Q(1)=0} V=R^3, W ={(x, y, z) elements of R^3|x+y inequality z}

Solution

V is the vector space of polynomials of upto 2^nd degree i.e. an element Q of V is of the form ax² + bx + c, where a, b, c are constants. W is a subset of V such that Q (1) = 0, i.e. a + b + c = 0.

Let Q₁ = a₁ x² + b₁ x + c₁ and Q₂ = a₂ x² + b₂ x + c₂ be two arbitrary elemts of W. Then Q₁ + Q₂ = a₁x² + b₁x + c₁ + a₂x²+ b₂x + c₂ = ( a₁ + a₂)x² + (b₁+b₂)x + (c₁ + c₂ ) . Also, ( a₁ + a₂) + (b₁+b₂)+ (c₁ + c₂ ) = ( a₁ + b₁ + c₁ ) + (a₂ + b₂ + c₂) = 0 + 0 = 0. Therefore, W is closed under addition. Further, if is an arbitrary scalar and Q = ax² + bx + c , an arbitrary element of W, then Q = (ax² + bx + c) = ax² + bx + c . Also, a + b + c = (a + b+ c) = (0) = 0, so that Q is closed under scalar multiplication. Also, when = 0, the zero vector i.e. a polynomial of zero degree is in W. Thus W is a linear subspace of V.