Linear Algebra (Math 220)
Assignment due Tuesday, April 1, 2008

1.  Preparation

Expect a quiz.

Relevant Reading:

• Lay §§ 4.7, 6.1, 6.2

• Hefferon §§ 3.V – 3.VI

2.  Exercises

1. Let ${\mathcal{P}}_d$ denote the vector space of polynomials of degree $d$ or less. If $f$ is an element of ${\mathcal{P}}_d$, let $I_f$ be the polynomial given by the formula $$I_f\left(x\right)={\int }_0^{x}f\text{.}$$

   1. Explain briefly why $I_f$ is linear.

   2. What is the kernel of $I_f\text{?}$

   3. In what set does the function $I_f$ takes its values (regarding ${\mathcal{P}}_d$ as its domain).

   4. What is the image of $I_f\text{?}$

2. What is the length of the line segment from the point $\left(2,-1,1\right)$ to the point $\left(4,-4,7\right)$ ?

3. What is the angle at the point $\left(0,1,-1\right)$ in the triangle whose vertices are that point, the point $\left(-1,3,1\right)\text{,}$ and the point $\left(3,7,-3\right)\text{?}$

4. Let $M$ be the $2\times 3$ matrix $$M=\left(\begin{array}{ccc}\hfill 3& \hfill 0& \hfill -1\\ \hfill 3& \hfill -2& \hfill 0\end{array}\right)\text{,}$$ and let $f$ be the linear function from ${\mathbf{R}}^3$ to ${\mathbf{R}}^2$ that is defined by $f\left(x\right)=Mx$. Find a basis of the kernel of $f$ consisting of vectors of length $1$.

5. Find a basis consisting of mutually perpendicular vectors for the plane in ${\mathbf{R}}^3$ defined by the linear equation $$2x-y+2z=0\text{.}$$