Math 260, Fall 2018, Assignment 9

The moving power of mathematical invention is not reasoning but the imagination.

- Augustus de Morgan

Read:

1. Section 3.3.

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

1. Linear relation (among a set of vectors $\vec{v}_1,\dots,\vec{v}_n$).
2. Trivial relation.
3. Linearly independent.
4. Basis (for a subspace of $\mathbb{R}^n$).
5. Dimension (of a subspace of $\mathbb{R}^n$).

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

1. Theorem relating redundancy, linear relations, and kernels.
2. Theorem relating the size of linearly independent sets (in a given subspace) to the size of spanning sets (for that subspace).
3. Theorem concerning the numbers of vectors in various bases for a given subspace.
4. Theorem concerning the existence of bases.

Solve the following problems:

1. Section 3.3, problems 5, 9, 15, 19, 23, 29, and 33.
2. Find a basis for $\mathbb{R}^n$.
3. Compute the dimension of $\mathbb{R}^n$.
4. Find a basis for the trivial subspace $\{\vec0\}$.
5. Compute the dimension of the trivial subspace.
6. In a previous assignment, we showed that any pair of two-dimensional planes in $\mathbb{R}^3$ must intersect in a line. Now, working in $\mathbb{R}^4$, find a pair of two-dimensional planes that intersect in a single point.

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