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HW10: Problem 7 Prev Up Next (1 pt) Let V be the vector space P3z of polynomials in with degree less than 3 and W be the subspace a. Find a nonzero polynomial p(z) in w p(z) b. Find a polynomial q(z) in V\W g(z) Note: You can earn partial credit on this problem. Preview Answers Submit Answers

Answer 1

(a). Consider .

Then, $p(x)=1.(7-9x-4x^2)+1.[6x^2-(6+x)] \in span \ W$ , and $p(x) \neq 0$ .

(b) Consider $q(x)=x$ . Then $q(x) \in V$ as is a polynomial in with degree less than .

We show that $q(x) \notin W$ .

To do this, assume that $q(x) \in W$ . Then for some scalars and .

$\Rightarrow x=x^2(6b-4a)+x(-9a-6b)+(7a-6b)$ .

Comparing coefficients of equal powers on both sides of te above equation, we have

.Solving the first and third equations, we get $b= \frac{2}{3}a$ and $b= \frac{7}{6}a$ respectively,

Therefore, $\frac{2}{3}a= \frac{7}{6}a \ \Rightarrow a=0$ . Then, from the firat equation, we get $b=0$ .

But, this gives us , which is not possible and therefore a contadiction to our assumtion that $q(x) \in W$ .

Thus, we have shown that $q(x) \in V$ and $q(x) \notin W$ . Hence $q(x) \in V\setminus W$ .

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