Matrix Theory Homework Question.(Picture attached) I would like to know the answ

Question

Matrix Theory Homework Question.(Picture attached)

I would like to know the answers for the question "1" (a)~(d). I am having a trouble understanding the question and how to start & how to answer these questions.

I would love a crystal clear "written answers".

Thanks for your help!

In each of the following you are to test whether the set with the operations given forms a real vector space. The candidate for vector addition is written as an ordinary plus sign, and multiplication is indicated by juxtaposition (writing one thing next to another). If the set with the operations is a vector space, you need only say so. If it is not, you must give one reason why not (a specific counterexample to one of the vector-space properties). Let V he the set of integers (positive, negative and zero), with x + y (as vectors) = x + y (ordinal addition) and alpha x (as a scalar times a vector) = alpha x (ordinary multiplication). Let V = {(x, y) R2 |x + 2y = 0} with addition and multiplication by scalars as in (a). Let V = {(x, y, z) R3 |2 = x2 + y2} with The question is the same as for 1. Let V be the set of all continuous functions f from R to R such that f(x) = f(x + 2 pi) for all x with

Explanation / Answer

(a)

NOT a vector space:
reason:

let (x,y) belongs to V

=>

x,y>=0

=>

-1(x,y) = (-x,-y) but -x,-y<=0 =>

(-x,-y) doesnt belong to V

=>
-1(x,y) doesnt belong to V

=>

V is not a vector space

(b)

NOT a vector space

reason

integer 1 belongs to V

but 0.5(1) = 0.5 doesnt belong to V

=>

V is not a vector space

(c)

Vector space

(d)

NOT a vecot space

(3,4,5), (12,5,13) belong to V

but

(3,4,5)+(12,5,13) = (15,9,18) doesnt belong to V
=>
V is not a vector space

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