Calculational Exercises
1. Show that the vectors , and are linearly independent in . Write as a linear combination of , and .
2. Consider the complex vector space and the list of vectors in , where
(a) Prove that
(b) Prove or disprove: is a basis for
3. Determine the dimension of each of the following subspaces of .
(a)
(b)
(c)
(d)
(e)
4. Determine the value of for which each list of vectors is linear dependent.
(a) as a subset of
(b) as a subset of
5. Consider the real vector space For each of the following five statements, provide either a proof or a counterexample.
(a)
(b)
(c) The list is linearly independent.
(d) Every list of four vectors , such that , is linearly independent.
(e) Let and be two linearly independent vectors in . Then, there exist vectors , such that is a basis for
Proof-Writing Exercises
1. Let be a vector space over and define , where for each
. Now suppose . Prove
2. Let be a vector space over , and suppose that the list of vectors spans , where each . Prove that the list
also spans
3. Let be a vector space over , and suppose that is a linearly independent list of vectors in . Given any such that
is a linearly dependent list of vectors in , prove that
4. Let be a finite-dimensional vector space over with for some . Prove that there are one-dimensional subspaces of such that
5. Let be a finite-dimensional vector space over , and suppose that is a subspace of for which Prove that
6. Let denote the vector space of all polynomials with degree less than or equal to and having coefficient over , and suppose that satisfy . Prove that is a linearly dependent list of vectors in
7. Let and be five-dimensional subspaces of . Prove that
8. Let be a finite-dimensional vector space over and suppose that are any subspaces of . Prove that
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