let H be the set of all polynomials of the form P(t)=a+bt^2 where a and b are in
ID: 1888244 • Letter: L
Question
let H be the set of all polynomials of the form P(t)=a+bt^2 where a and b are in R and b>a. 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 inder vector addition C: closed under multiplication by scalars A) His not a vector space; does not contain zero vector B)H is not a vector space; not closed under multiplication by scalars and does not contain zero vector C) H is not a vector space; not closed under vector addition D) H is not a vector space; not closed under multiplication by scalars.Explanation / Answer
It fails properties A and C (if non-positive scalars are allowed), but passes B. A: The zero vector in this case is 0 = 0 + 0t2 (because adding this vector to any vector v yields v; the zeroes are added coefficient-wise, meaning that 0 in R should correspond to the zero vector in H). However, 0 > 0 is false, so the zero vector is not in H. Therefore, H does not contain the zero vector. B: Let's say we have vectors u = a + bt2 and v = c + dt2. u + v = (a + c) + (b + d)t2, which is of the form a vector in H must be in. Now, because b > a and d > c, b+d > a+c, so u + v is indeed in H. Therefore, H is closed under vector addition. C: Is it true that cu is in H for any vector u in H? If we let u = a + bt2, then cu = ca + cbt2. If cb > ca, then cu is in H. However, if c = 0, then cb = 0 and ca = 0, so cb > ca is false. Also, if c < 0, cb ca is false again. In either case, because u is not closed under multiplication by scalars, H is not a vector space.Related Questions
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