Math 260, Fall 2019, 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. Basis (for a subspace $S$).
2. Dimension (of a subspace $S$).
3. Coordinates (of a vector $\vec{v}$, with respect to the basis $\mathcal{B}=(\vec{v}_1,\dots,\vec{v}_k)$).
4. $[\vec{v}]_{\mathcal{B}}$ (the coordinate vector of $\vec{v}$ with respect to the basis $\mathcal{B}=(\vec{v}_1,\dots,\vec{v}_k)$).

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

1. Theorem relating the size of a linearly independent set (in a given subspace $S$) to the size of a spanning set (for the same subspace $S$).
2. Theorem relating the sizes of two bases for the same subspace $S$.
3. Theorem concerning the refinement of spanning sets to bases.
4. Theorem concerning the extension of linearly independent sets to bases.
5. Theorem relating the dimensions of $V$ and $W$, when $V\subseteq W$.
6. Theorem relating $V$ and $W$, when $V\subseteq W$ and $\dim V=\dim W$.
7. Theorem concerning the unique expansion of $\vec{v}\in S$ as a linear combination of the vectors in a basis $\mathcal{B}=(\vec{v}_1,\dots,\vec{v}_k)$ for $S$.

Solve the following problems:

1. Section 3.3, problems 1, 5, 7, 9, 15, 25, 27, 29, 30, and 33.
2. Section 3.4, problems 1 and 3.
3. Suppose $S$ is a $d$-dimensional subspace of $\mathbb{R}^n$. Is it possible to span $S$ with fewer than $d$ vectors? Carefully prove your answer.
4. Suppose $S$ is a $d$-dimensional subspace of $\mathbb{R}^n$. Is it possible to make a list of more than $d$ linearly independent vectors in $S$? Carefully prove your answer.

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