B only A and B A only QUESTION5 Solve the problem. Let H be the set of all point

Question

B only A and B A only QUESTION5 Solve the problem. Let H be the set of all points of the form (s, s-1). Determine whether H is a vector space. If it is not a vector space, determine which of the following properties it fails to satisfy. A: Contains zero vector B: Closed under vector addition C: Closed under multiplication by scalars His not a vector space, does not contain zero vector H is not a vector space; not closed under vector addition H is a vector space. H is not a vector space: fails to satisfy all three properties Click Save and Submit to save and submit. Click Save All Answers to save all answers Save All A

Explanation / Answer

H does not contain the zero vector since the zero vector (0,0) is not of the form (s,s-1)

If for some s, (0,0)=(s,s-1) then, s=0 and s=1 which does not hold

Hence H does not contain the zero vector

H is not even closed under vector addition.

Take two points in H, (s,s-1) and (t,t-1)

Then their sum is, (s+t, s+t-2) which is not of the required form (s+t, s+t-1)

Also H is not closed under scalar multiplication

Take c be any scalar

Then, c(s,s-1) = (cs, cs-c) which is equal to (cs,cs-1) only if c=1

Hence H fails to satisfy all the axioms to be a vector space.

Hence, option (4) is correct

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