?Q6 spring 09 can you help me with this Quesrtion for my Differentail Equation,

Question

?Q6 spring 09

can you help me with this Quesrtion for my Differentail Equation, and Linear Algebra class.

FinalExamSpring20092243 34324.pdf (page 4 of 17) Scale document up 6. Let V be the set of all polynomials of degree 3 or less, let Wi be the set of all polynomials of the form ats +b(t 1)2, let W2 be the set of all polynomials of the form t2 +at b, let Ws be the set of all polynomials of the form a2t2, where a and b can be any real numbers. Then, (A) W is a subspace of V, but W2, Ws are not. (B) W2 is a subspace of V, but Wi, Ws are not. (C) Ws is a subspace of V, but W1, W2 are not. (D) Wi.W2.Ws are all subspaces of V (E) None of the above is correct.

Explanation / Answer

(A) W1 is a subset of V,but W2,W3 are not.

To be a subspace of V which is the set of all polynomials of degree 3 or less,the subset needs to be of order 3(  highest degree of its individual terms with non-zero coefficients should be 3).

polynomial should satisfy two ingredients like scalar multiplication(multiplication of a scalar quantity with the polynoimal results the polynomial having same degree ) and addition(addition of two subspace polynomials gives a polynomial which becomes a polynomial of same degree and subspace of original set of polynomials).

in the above question W1 satisfy all these three conditions ,but W2 & W3 dont ,so W1 is a subset of V but w2 ,W3 are not.

only restriction is : coffiecient a and b should be nonzero.

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