Math 260, Fall 2017, Assignment 7

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

- Lord Kelvin

Read:

1. Section 3.2.
2. Section 3.3.

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

1. Redundant vector.
2. Linearly independent (set of vectors).
3. Basis (for a subspace).
4. Linear relation (among a set of vectors).
5. Trivial relation.
6. Dimension (of a subspace).

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

1. Theorem relating redundancies to relations.
2. Theorem relating relations to kernels.
3. Theorem describing two conditions equivalent to linear independence.

Carefully describe the following algorithms:

1. Algorithm to detect redundancies in a set of vectors.
2. Algorithm to produce a basis for the image of a matrix.
3. Algorithm to produce a basis for the kernel of a matrix.
4. Algorithm to find all linear relations among a set of vectors.

Solve the following problems:

1. Section 3.2, problems 11, 15, 17, 18, 22, 23, 24, 27, 29, 33, 34, and 42.
2. Section 3.3, problems 1, 2, 15, 21, 23, 29, and 33.

--------------------End of assignment--------------------