Math 260, Fall 2017, Assignment 6

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

- Goethe

Read:

1. Section 3.1.
2. Section 3.2, through page 124.

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

1. Domain (of a transformation).
2. Codomain (of a transformation).
3. Image (of a transformation).
4. Kernel (of a linear transformation).
5. Linear subspace (of $\mathbb{R}^n$).
6. Linear combination (of vectors $\vec{v}_1,\dots,\vec{v}_d$).
7. Span (of a set of vectors).

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

1. Theorem relating images of linear transformations to linear subspaces (a.k.a. "some properties of the image").
2. Theorem relating kernels of linear transformations to linear subspaces (a.k.a. "some properties of the kernel").
3. Theorem describing the image of a linear transformation as the span of a certain set of vectors.

Solve the following problems:

1. Section 3.1, problems 1, 3, 5, 9, 13, 14, 15, 16, 17, 19, 23, 25, 30, and 33 (hint for problems 1-13: the equation $A\vec{x}=\vec{0}$ is equivalent to a certain linear system, which you can solve by Gauss-Jordan elimination.)
2. Section 3.2, problems 1, 2, 3, and 6.

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