Math 260, Spring 2017, 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. (Linear) subspace (of $\mathbb{R}^n$).
2. Redundant vector (in a list $\vec{v}_1,\dots,\vec{v}_k$).
3. Linear relation (among $\vec{v}_1,\dots,\vec{v}_k$).
4. Trivial relation.
5. Linearly independent (vectors).
6. Basis (for a subspace $V$ of $\mathbb{R}^n$).
7. Coordinates (of a vector $\vec{v}$ with respect to a basis $\mathcal{B}$).
8. Dimension (of a subspace).

Carefully describe the following algorithms:

1. Algorithm to detect redundancies in a list of vectors.

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

1. Theorem relating spans to redundancies.
2. Theorem relating linear relations to redundancies.
3. Theorem relating linear relations to kernels.
4. Theorem giving three conditions, each of which is equivalent to linear independence.
5. Theorem concerning unique expansion of a given vector on a given basis.

Solve the following problems:

1. Section 3.2, problems 1, 2, 3, 4, 11, 13, 15, 18, 21, 25, 27, 29, and 34.

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