Homework 4: Subspace, Linear Combination, Span

1. Section 4.2, #3(a), #5(a)(b), #10(b)(c), 12(a)(b), 14.
2. Give an example of a nonempty subset of \(M_{2,3}\) which is not a subspace of \(M_{2,3}\).
3. Sketch the following: (a) \(\sspan\{\vtwo{1}{2}, \vtwo{2}{3}\}\), (b) \(\sspan\{\vtwo{1}{2}, \vtwo{2}{4}\}\), (c) \(\sspan\{\vtwo{0}{0}\}\),
   (d) \(\sspan\left\{\vthree{1}{2}{3}\right\}\), (e) \(\sspan\left\{\vthree{1}{2}{0}, \vthree{1}{2}{3}\right\}\).
4. Consider the set \( W \) of all vectors in \(\bR^4\) of the form \(\vfour{\pe a}{\pe b}{-2b}{\pe a}\). Verbally, we can describe \(W\) as the set of all vectors in \(\bR^4\) whose first and fourth components are equal, and whose third component is -2 times its second component. Using set-builder notation, we can express \(W\) as \[ W = \left\{ \vfour{\pe a}{\pe b}{-2b}{\pe a} : a, b \text{ are real numbers}\right\} \]
   1. Prove that \(W\) is a subspace of \(\bR^4\) using Theorem 1.2 from the study guide.
   2. Prove that \(W\) is a subspace of \(\bR^4\) using Theorem 1.3 from the study guide.
5. Let \(\bu, \bv\), \(\bw\), \(\bx\), \(\by\), and \(\bzz\) be vectors in a vector space \(V\). Using only the definition of \(\sspan\) (and no theorems), prove that if \(\bzz\) is in \(\sspan\{\bx,\by\}\), and \(\bx\) and \(\by\) are in \(\sspan\{\bu,\bv,\bw\}\), then \(\bzz\) is in \(\sspan\{\bu,\bv,\bw\}\).
6. (Optional, Putnam Competition) Let \(S\) be a set and let \(\circ\) be a binary operation on \(S\) satisfying the two laws \begin{align*} &x \circ x = x \text{ for all \(x\) in \(S\), and}\\ & (x \circ y)\circ z = (y\circ z)\circ x \text{ for all \(x,y,z\) in \(S\)}. \end{align*} Show that \(\circ\) is associative and commutative.