Linear Algebra (Math 220)
Assignment due Tuesday, March 4

1.  Preparation

Expect a quiz.

Relevant Reading:

• Lay § 1.7 and § 4.3

• Hefferon § 3.II – 3.III

• Matthews §§ 3.3 – 3.4

2.  Exercises

1. Which sets of column indices correspond to maximal linearly independent sets of columns in the following matrices?

   1. $$\left(\begin{array}{cc}\hfill 18& \hfill -42\\ \hfill -15& \hfill 35\end{array}\right)$$

   2. $$\left(\begin{array}{ccc}\hfill 3& \hfill -2& \hfill 4\\ \hfill -1& \hfill 5& \hfill 2\\ \hfill 5& \hfill -12& \hfill 0\end{array}\right)$$

   3. $$\left(\begin{array}{ccc}\hfill 1& \hfill 2& \hfill 3\\ \hfill 4& \hfill 5& \hfill 6\\ \hfill 7& \hfill 8& \hfill 9\end{array}\right)$$

2. Which sets of row indices correspond to maximal linearly independent sets of rows in the following matrices?

   1. $$\left(\begin{array}{cccc}\hfill 1& \hfill 2& \hfill -4& \hfill 7\\ \hfill -2& \hfill -1& \hfill -1& \hfill -8\\ \hfill -1& \hfill -4& \hfill -14& \hfill 5\\ \hfill 5& \hfill 7& \hfill -11& \hfill 29\end{array}\right)$$

   2. $$\left(\begin{array}{cccc}\hfill 1& \hfill 2& \hfill -4& \hfill 7\\ \hfill -2& \hfill -1& \hfill -1& \hfill -8\\ \hfill -3& \hfill -6& \hfill 12& \hfill -21\\ \hfill 5& \hfill 7& \hfill -11& \hfill 29\end{array}\right)$$

3. In the vector space of all differentiable functions of the real variable $t$ find a maximal linearly independent subset in the subspace spanned by the 4 functions $\cos t$, $\sin t$, ${\cos }^{2}t$, and ${\sin }^{2}t$.