Math 260, Fall 2015, Assignment 7

Mathematicians are like Frenchmen: whatever you say to them they translate into their own language, and forthwith it is something entirely different.

- Goethe

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

1. Redundant vector (in a list $\vec{v}_1,\dots,\vec{v}_k$).
2. Linear relation (among $\vec{v}_1,\dots,\vec{v}_k$).
3. Trivial relation.
4. Linearly independent set.
5. Basis (for some subspace $S\subseteq\mathbb{R}^n$).

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

1. Theorem relating redundancy to relations.

Carefully describe the following algorithms:

1. Algorithm to find all redundant vectors in a list.
2. Algorithm to compute a basis for the image of a linear transformation.

Solve the following problems:

1. Section 3.2, problems 8, 9, 11, 13, 15, 19, 25, 29, and 33.

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