ANSWER BOTH QUESTIONS ACCRUATELY AND CORRECTLY FOR A GOOD RATING Q5. Consider any vector space V. Let x,y,z be elements of V and a,b be scalars. (a) Prove: If x+y=x+z then y=z. (b) Prove: (a+b)(x+y)=ax+bx+ay+by. Q6. Consider polynomials with complex coeffici

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ANSWER BOTH QUESTIONS ACCRUATELY AND CORRECTLY FOR A GOOD RATING Q5. Consider any vector space

V

. Let

x,y,z

be elements of

V

and

a,b

be scalars. (a) Prove: If

x+y=x+z

then

y=z

. (b) Prove:

(a+b)(x+y)=ax+bx+ay+by

. Q6. Consider polynomials with complex coefficients and degree at most

n

.

P_(n)(C)={a_(0)+a_(1)x+a_(2)x^(2)+cdots+a_(n)x^(n):a_(i)inC}

Prove that

P_(n)(C)

is a vector space. (Hint: summation notation will simplify your argument.)

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