Math 260, Spring 2017, Assignment 9

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. Nullity (of a linear transformation).
2. Coordinate vector (of a vector $\vec{v}$ with respect to a basis $\mathcal{B}=(\vec{v}_1,\dots,\vec{v}_d)$)

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

1. Lemma relating the size of a spanning set to the size of a linearly independent set.
2. Theorem relating the sizes of two bases for the same subspace.
3. Theorem concerning extension of linearly independent sets to bases.
4. Theorem concerning refinement of spanning sets to bases.
5. Rank-Nullity Theorem.

Carefully describe the following algorithms:

1. Algorithm to produce a basis for an image.
2. Algorithm to produce a basis for a kernel.
3. Algorithm to compute the coordinates of a vector $\vec{v}$ with respect to a basis $\mathcal{B}=(\vec{v}_1,\dots,\vec{v}_d)$.

Solve the following problems:

1. Section 3.3, problems 1, 3, 7, 19, 23, 25, 27, 29, 33, 35, 37, and 38.
2. Section 3.4, problems 1, 3, 7, 13, 17, 48, and 49.

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