Linear Algebra (Math 220)
Assignment due Thursday, April 17

1.  Preparation

Expect a quiz.

Relevant Reading:

Course notes on “change of basis” (also available as PDF)

Lay § 4.7

Hefferon § 3.V

2.  Exercises

1. Let ${\mathcal{P}}_2$ denote the vector space of polynomials of degree $2$ or less. If $f$ is an element of ${\mathcal{P}}_2$, let $T_f$ be the polynomial given by the formula $$T_f\left(x\right)=\frac{d}{dx}\left(xf\left(x\right)\right)\text{.}$$

   1. Show that the function $T$ that is defined by $$T\left(f\right)=T_f$$ is a linear map from ${\mathcal{P}}_2$ to ${\mathcal{P}}_2$.

   2. What is the dimension of ${\mathcal{P}}_2$ ?

   3. Find a basis of the kernel of $T$.

   4. Find a basis of the image of $T$.

2. Let $f$ be the linear function from ${\mathbf{R}}^3$ to ${\mathbf{R}}^3$ that has the matrix $$D=\left(\begin{array}{ccc}\hfill 2& \hfill 0& \hfill 0\\ \hfill 0& \hfill 1& \hfill 0\\ \hfill 0& \hfill 0& \hfill 3\end{array}\right)$$ relative to the basis of ${\mathbf{R}}^3$ given by the columns of the matrix $$\left(\begin{array}{ccc}\hfill 3& \hfill 6& \hfill 2\\ \hfill 2& \hfill -3& \hfill 6\\ \hfill 6& \hfill -2& \hfill -3\end{array}\right)\text{.}$$ Find the matrix of $f$ relative to the standard basis of ${\mathbf{R}}^3$.