math54 - quiz 4 - solns

# math54 - quiz 4 - solns - integer need not be an integer...

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Math 54 quiz 5 Solutions 1. Mark each of the following statements TRUE or FALSE. No justiﬁca- tion is needed. (1 point each) (a) The set { 0 V } is a subspace of V for any vector space V . TRUE (b) V is a subspace of itself for any vector space V . TRUE (c) R 3 is a subspace of R 4 . FALSE (it’s not a subset!) 2. Find an example of each of the following. Give a short justiﬁcation of why your example works. (2 points each) (a) A vector space V and a subset S such that Span( S ) = V , but S is linearly dependent. Take for example V = R 1 , and S = { [1] , [0] } . Then the ele- ment [1] in S spans all of V , and because [0] is in S , it must be linearly dependent. For example, we have the nontrivial linear combination 0[1] + 2[0] = [0] producing the zero vector. (b) A subset of the vector space R 1 that is closed under addition and contains the zero vector (which in this cas is the number 0) but is not closed under scalar multiplication. Hint: what are some well known subsets of R ? The integers is one possibility. The sum of any two integers is an integer, zero is an integer, but a (real number) multiple of an

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Unformatted text preview: integer need not be an integer. The rational numbers also works, as does the natural numbers. 1 3. (3 points)Let V be a vector space and let U and W be subspaces of V . Deﬁne U + W = { u + w ∈ V | u ∈ U, w ∈ W } . Show that U + W is a subspace of V . First we check that is in U + V . This follows from taking the element + in U + V , noting that is in both U and in V . Now to check the closure properties, we need two write down two arbitrary elements of U + V . We will write these as u 1 + v 1 and u 2 + v 2 . Also, take a scalar a ∈ R . Then ( u 1 + v 1 ) + ( u 2 + v 2 ) = ( u 1 + u 2 ) + ( v 1 + v 2 ) . The vector on the right is in U + V because U and V are closed under addition (so we have written it as something in U + something in V ). Next we have a ( u 1 + v 1 ) = a u 1 + a v 1 . This is in U + V because U and V are closed under scalar multiplica-tion. 2...
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math54 - quiz 4 - solns - integer need not be an integer...

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