Math 333 - Practice Exam
(Note that the exam will NOT be this long.)
1 Definitions
1. (0 points) Let Ube a subset of a vector space V. Let S={v1, v2, . . . , vn}be
another subset of V.
(a) Define “Uis a subspace of V”.
(b) Define “Sis linearly independent”.
(c) Define “Sgenerates V”.
2 Vector Spaces and Subspaces
2. (0 points)
(a) Give three examples of 4-dimensional vector spaces.
(b) Give one example of an infinite dimensional vector space.
(c) Give an example of a zero-dimensional vector space.
3. (0 points) Let S1and S2be subspaces of a vector space V. Prove that the union
S1∪S2is a subspace of Vif and only if one is contained in the other (that is, either
S1⊆S2or S2⊆S1.)
3 Linear Independence, Generating Sets, and Bases
4. (0 points) Let S={x2+ 3x, x −2}be a subset of P2(R).
(a) Explain why Sis not a basis of P2(R).
(b) Is 1
3x2+ 2 in span(S)? Explain.
(c) Is 2x2+ 5x+ 4 in span(S)? Explain.
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