Linear Algebra (Math 220)
Assignment due Thursday, March 13

1.  Preparation

Expect a quiz.

Bring Questions in preparation for the Midterm Test.

2.  Exercises

1. When $M$ is an $m\times n$ matrix, the phrase “corresponding linear function” will denote the linear function $${\mathbf{R}}^n\stackrel{f_M}{\to }{\mathbf{R}}^m$$ defined by $$f_M\left(x\right)=Mx\text{for}x\text{in}{\mathbf{R}}^n\text{.}$$ In the case $m=2,n=3$ $$M=\left(\begin{array}{ccc}\hfill 3& \hfill 6& \hfill 0\\ \hfill 2& \hfill 4& \hfill 1\end{array}\right)$$ compute each of the following items both for (i) $M$ itself and for (ii) its reduced row echelon form:

   1. The set of linear combinations of the columns.

   2. The set of linear combinations of the rows.

   3. The set of linear relations among the columns.

   4. The set of linear relations among the rows.

   5. The kernel of the corresponding linear function.

   6. The image of the corresponding linear function.

2. Let $Q_3$ be the $4$-dimensional vector space consisting of all polynomials of degree $3$ or less, and let $$\mathbf{v}=\left\{1,t,t^2,t^3\right\}$$ be the familiar basis of $Q_3$. Let $Q_3\stackrel{\varphi }{\to }{Q}_3$ be the linear map that is defined by $$\varphi \left(P\right)=P^{\prime \prime }+3P^{\prime }+2P\text{,}$$ where $P^{\prime }$ and $P^{\prime \prime }$ denote the first and second derivatives of $P$. Find the matrix of $\varphi$ with respect to the basis $\mathbf{v}$, i.e., find the $4\times 4$ matrix $R$ that appears in the transport diagram $$\begin{array}{ccc}\hfill Q_3\hfill & \hfill \stackrel{\varphi }{\to }\hfill & \hfill Q_3\hfill \\ \hfill {\alpha }_{\mathbf{v}}\uparrow \hfill & \hfill \hfill & \hfill \uparrow {\alpha }_{\mathbf{v}}\hfill \\ \hfill {\mathbf{R}}^4\hfill & \hfill \stackrel{f_M}{\to }\hfill & \hfill {\mathbf{R}}^4\hfill \end{array}\text{.}$$