How do you determine whether a set equipped with the usual addition and scalar m

Question

How do you determine whether a set equipped with the usual addition and scalar multiplication are real vector spaces?

1. The set of differentiable functions f : ??? such that f?(0) = f?(1).

2. The set of real-valued functions on ? such that lim f(x) = 1

x??

3. The set of polynomials with real coefficients of degree exactly n.

4. {(x,y)??2: sin(x + y) = 0}

5. Set of polynomials with real coefficients that are missing the term of degree 13.

6. {(x,y)??2: x +?y + 1? 0}

7. The set of functions f : ???that are increasing.

(m) The set of functions f : ??? that are zero at x =132 .

(n) The set of functions f : ??? such that f(2) = 13

Explanation / Answer

Look up the def of vector spaves 1. Yes 2. No, its not closed under scalar mult 3. No, We can add 2 poly of deg n and obtain another with deg(n-1) 4. Yes 5. Yes. 6. No not closed 7. Yes m. yes n. No not closed under addition message me if you have any doubts about the answer

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