Math 260, Fall 2017, Assignment 8

Mathematical proofs, like diamonds, are hard as well as clear, and will be touched with nothing but strict reasoning.

- John Locke, Second Reply to the Bishop of Worcester

Carefully define the following terms, then give one example and one non-example of each:

1. Coordinates (of a vector $\vec{x}$ with respect to a basis $\mathcal{B}=\{\vec{v}_1,\dots,\vec{v_k}\}$).

Carefully state the following theorems (you need not prove them):

1. Theorem relating the size of a linearly independent set in a subspace $S$ to that of a spanning set in the same subspace.
2. Invariance of dimension.
3. Theorem concerning the existence of bases.
4. Theorem concerning the extension of linearly independent sets to bases.
5. Theorem concerning the refinement of spanning sets to bases.
6. Theorem relating the dimensions of "nested" subspaces $S\subseteq T$.
7. Theorem concerning nested subspaces $S\subseteq T$ whose dimensions are equal.

Solve the following problems:

1. Section 3.3, problem 27, 34, 35, 38, and 64.

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