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{\bf{Extra credit assignment: review problems. Part 1.}}
Part 2 will be posted soon.
The whole assignment is worth $5\%$ and can be done instead of the computer project, or in addition to it for extra credit.
%\begin{enumerate}
{1. \bf Lines and planes in $\R^3$ and in higher dimension.}
Remember from multivariable calculus that a line in $\R^3$ can be given by a parametric equation, or by ``symmetric equation''
$$ \frac{x-x_0}a = \frac{y-y_0}b = \frac{z-z_0}c$$.
Note that the ``symmetric equation'' is, in fact, \emph{two} equations, and together they represent your line as an intersection of two planes, one vertical, one horizontal.
A plane in $\R^3$ (if it contains the origin) is a linear subspace of \emph{co-dimension} $1$ (so it is given by a single equation on $x,y,z$). Note that the way the equation for a plane is introduced in multivariable calculus uses the dot product: it is given via a normal vector $(a, b,c)$ (then the equation for a plane containing $0$ is $$(x,y,z)\cdot (a,b,c)=0,$$ which gives
$ax+by+cz=0$). However, in the end we do not need the dot product to think of a plane, it is just a linear subspace of $\R^3$ defined by one linear relation on its coordinates (in other words, it is a space of solutions of a single linear equation).
Before proceeding, make sure you understand these statements.
In this problem, solve all the parts \emph{without using cross product} (in $\R^3$, the cross product simplifies some of the calculations, but it does not generalize to higher-dimensional spaces; here the goal is to illustrate what happens in $\R^n$, so we do not want to be distracted by the cross product).
\begin{enumerate}[(a)]
\item Write down a general vector form of a parametric equation for a line in $\R^3$ that passes through the origin (namely, a $1$-dimensional linear subspace of $\R^3$).
\item Write down a general form of a parametric equation for a line in $\R^n$ that passes through the origin (namely, a $1$-dimensional linear subspace of $\R^n$).
\item Write down a general form of the ``symmetric equation'' for a line in $\R^4$. How many equations does it really consist of?
\item If you take a ``system'' of linear equations that consists of a single equation $x+3y+2z=0$ and ``solve'' it using the methods from our course, what do you get?
(I claim that you should get a \emph{parametric} equation for your plane, and it will use two parametrs).
Interpret this solution in terms of a basis. Namely, find a basis for the plane $x+3y+2z=0$.
\item How many equations do you need to define a plane in $\R^4$? Why? Can you use rank-nullity theorem to give a rigorous proof?
\item Find a basis for the plane $W$ in $\R^4$ defined by the equations
$$\begin{aligned}
&x+2y+3z+w=0\\
&y-z-5w=0
\end{aligned}
$$
\item Interpret your result in the previous part as `` The kernel of the linear map ... is the linear subspace $W$ with basis ..."
(Fill in the blanks in the above sentence).
\item Write down a linear equation that defines the plane (call it $W$) with basis $\{(1,1,1), (2,3,1)\}$ in $\R^3$ (without using cross product!)
I claim that I can interpret your result in the following language:
``Find a \emph{linear functional} $f:\R^3\to \R$ such that $\Ker(f)=W$." Write down this linear functional.
\item Write down a system of linear equations in $\R^4$ that define the plane $W$ with basis $\{(1, 1,1,1), (2,3,0,1)\}$ and interpret your result as ``$W=\Ker(A)$, where $A$ is a linear map ....''.
\item Let $f_1:\R^3\to \R$ be the linear functional $f_1(x,y,z)= x+2y+3z$, and let $f_2:\R^3\to \R$ be the linear functional $f_2(x,y,z)=x-y-z$.
Give a complete description of $\Ker(f_1)\cap \Ker(f_2)$ in as many ways as you can, as precisely as you can.
\item How many linear equations do you need to define a plane in $\R^n$? Why?
\item Decide whether the line with parametric equation $(x,y,z)=(t,2t, 3t)$ is contained in the plane given by the equation $x+y-z=0$.
Now I am going to ask this question in a different way:
``Let $v=(1,2,3)$, and let $w_1$ and $w_2$ be a basis for the plane $x+y-z=0$. Is the collection of vectors $\{v, w_1, w_2\}$ linearly independent? ''
\item Let $W_1$ be the subspace of $\R^5$ defined by the equation $x_1+x_2+x_3+x_4+x_5=0$, and let $W_2$ be defined by the equations
$x_1-x_2+x_3=0$ and $x_2-2x_3 +x_5=0$. Find the dimensions of $W_1$ and $W_2$, and find a basis for $W_1\cap W_2$.
\end{enumerate}
{\bf II. Linear maps and their matrices.}
The fundamental idea is that if you have a basis in $V$ and a basis in $W$, you can define a linear map from $V$ to $W$ using a matrix. In the matrix, the columns are the coordinates of the images (in $W$) of the basis vectors of $V$. In doing this, you have complete freedom as to where your map sends the basis vectors.
\begin{enumerate}
\item Let $P_n$ be the space of polynomials of degree not greater than $n$, and let $D:P_n\to P_n$ be the linear map given by $D(p)=p'$ (the derivative of $p$). Write down the matrix for $D$ with respect to the basis $\{1, x, x^2, \dots, x^n\}$. What is the size of this matrix?
Find the kernel and image of $D$. Find the eigenvalues and eigenvectors of $D$ in $P_n$.
Prove that $D^n=0$ (so $D$ is a so-called \emph{nilpotent} linear operator).
\item Curtis, Section 13, Problem 1 (p.107)
\item Curtis, Section 13, Problem 2 (p.107)
\item Explain whether there exists a linear transformation $T:\R^4\to \R^2$ such that $T(0,1,1,1)= (2,0)$, $T(1,2,1,1) = (1,2)$,
$T(1,1,1,2) = (3,1)$, $T(2, 1,0, 1) = (2,3)$. (This is problem 11 in Curtis).
\item
Let $C^\infty((0,1))$ be the space of all infinitely-differentiable functions on the interval $(0,1)$ (it is infinite-dimensional).
\begin{enumerate}[(a)]
\item Prove that $e^{\lambda x}$ is an eigenvector of the differentiation operator $D: C^\infty((0,1)) \to C^\infty((0,1))$ defined by $D(f)=f'$.
Conclude that the functions $e^{\lambda x}$ form a linearly independent set of vectors in $C^\infty((0,1))$.
\item Consider the linear subspace $W$ of $C^\infty((0,1))$ spanned by $\{e^x, e^{2x}, e^{3x}\}$ (by the previous part, you know that they are linearly independent, so $W$ is $3$-dimensional).
Let $A_1:W\to \R^3$ be defined by $A_1(f)= (f(1), 3f'(2), f''(3))$.
Find the matrix for the linear map $A_1$ with respect to the given basis of $W$ and the standard basis of $\R^3$.
\end{enumerate}
\end{enumerate}
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