Math 260, Spring 2017, Assignment 4

We admit, in geometry, not only infinite magnitudes, that is to say, magnitudes greater than any assignable magnitude, but infinite magnitudes infinitely greater, the one than the other. This astonishes our dimension of brains, which is only about six inches long, five broad, and six in depth, in the largest heads.

- Voltaire

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

1. $n$-component column vector.
2. $n$-component row vector.
3. $\mathbb{R}^n$.
4. Standard basis vectors (in $\mathbb{R}^n$).
5. Sum (of two column vectors, or two row vectors, or two matrices).
6. Scalar multiple (of a column vector, or a row vector, or a matrix).
7. Dot product (of two vectors).
8. Length (of a vector).
9. Angle (between two vectors).
10. Orthogonal (vectors).
11. Product (of two matrices).
12. Matrix form (of a linear system).

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

1. Distributive law for dot products.
2. Geometric interpretation of the dot product.
3. Theorem regarding commutativity of matrix multiplication.
4. Theorem regarding associativity of matrix multiplication.

Solve the following problems:

1. Section 1.3, problems 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 27, 28, 34, and 35.
2. Section 2.3, problems 1, 3, 5, 7, 10, 11, and 13.
3. Section 5.1, problems 1, 3, 5, 6, and 9.

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