Which of the following sets constitute a vector space? The answer must be in one

ID: 2964376 • Letter: W

Question

Which of the following sets constitute a vector space? The answer must be in one of the two forms:

V is a vector space because it is closed under addition and multiplication by scalars

V is not vector space because.. (give reason)

a) The set V2 all of all vectors on the line y = x in the xy-plane:

V2 = {all {a},{a}}1x2

Illustrate your answer with a picture. If the answer is negative, also give a second reason (with an example)

b) The set V3 of all vectors of the length one in a plane:

V3 = {all {x},{y}, where x2+y2=1}

Illustrate your answer with a picture. If the answer is negative, also give a second reason (with an example)

Please show all work! Thanks!

a) the type of vector on y=x is (a,a) general form

closed under addition say two vectors (a,a) and (b,b)

sum is (a+b,a+b) this satisfies the line y=x => true

closed under multiplication by scalar

say vector = (a,a) with scalar k = (ka,ka) this satisfies the line y=x

V2 is a vector space

b) v3 is not a vector space as (-1/2^(1/2) ,- 1/2^(1/2)) and (1/2^(1/2),1/2^(1/2))

sum is (0,0) which is not on the circle => it is not closed on addition

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