Linear algebra Can anyone shows me how to do the problem and show work? Let H be

Question

Linear algebra

Can anyone shows me how to do the problem and show work?

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. Contains zero vector Closed under vector addition Closed under multiplication by scalars if the set W is a vector space, find a basis of W and state the dimension of W. Otherwise, state that W is not a vector space. W is the set of all vectors of the form [a + 5b 6b 6a - b -a], where a and b are arbitrary real numbers. Let W = [a + 2b + 2d c + d -3a - 6b + 4c - 2d -c - d]: a, b, c, d in R}

Explanation / Answer

H is not a vector space as it does not contain the zero vector.

For ascertaining whether W is a vector space, let us first find out whether v1 , v 2, v3 and v4 are linearly independent where v1 = a + 5b , v2 = 6b, v3 = 6a -b and v4 = -a. It can be seen that v1 = - v4 + 5/6 v2 , v2 = ( -6) [ v3 + 6 v4], v3 = - 6v4 -1/6 v2 and v4 = - v1 + 5/6 v2. Since, v1, v2 , v3 and v4 are linear combinations of one-another, W is not a vector space.

Similarly, when v1 = a + 2b + 2d, v2 = c+d, v3 = - 3a- 6b + 4c - 2d and v4 = - c- d, we can see that v4 = - v2 . Thus the second W is also not a vector space.

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