Math 260, Fall 2015, Assignment 8

Do not imagine that mathematics is hard and crabbed and repulsive to commmon sense. It is merely the etherealization of common sense.

- Lord Kelvin

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

1. Dimension (of a subspace $V$ of $\mathbb{R}^n$).
2. Coordinates (of a vector $\vec{v}$ with respect to an ordered basis $\mathcal{B}$).

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

1. Lemma comparing the sizes of linearly independent sets with spanning sets.
2. Theorem concerning the sizes of two bases for the same subspace.
3. Theorem concerning the extension of linearly independent sets to bases.
4. Theorem concerning the refinement of spanning sets to bases.
5. Two theorems concerning the dimensions of nested subspaces.
6. Rank-Nullity Theorem.
7. Theorem concerning the uniqueness of coordinates.

Carefully describe the following algorithms:

1. Algorithm to compute a basis for the kernel of a linear transformation.

Solve the following problems:

1. Section 3.3, problems 3, 5, 7, 23, 27, 29, 31, and 33.

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