Homework 5.tex

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{\Large \noindent Math 212\hfill Common Errors in Homework \# 5}

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\begin{enumerate}
  \item Do not be sloppy: staple your homework and do not turn in paper that has been ripped out of a notebook without removing the tear-away frills on the side.
  \item Write complete sentences. Everything you write should be able to be read from top to bottom. Equations are sentences too, so make sure to use punctuation properly! Don't use symbols like $\therefore$ or $\implies$ in a sentence mostly words.
  \item The symbol $\to$ should only be used when discussing maps. For example, the function $f:\R^4\to P(\R^4)$ cannot be onto.
  \item The implication symbol is $\implies$, not $\to$.
  \item Do not turn in rough drafts! Check your work and write your solutions neatly and clearly before turning them in. Make sure that every word or symbol you put on the paper is necessary. Do NOT include work (as in examples) that you did to think through the problem unless it is necessary to the solution. Nothing should be crossed out or out of order.
  \item Think about the easiest and simplest way to do something. Don't do extra and unnecessary work.
  \item Do not use the word `it' unless you are 100\% sure that it is clear what \textit{it} is referring to. It is best practice to eliminate `it' (and other words like `it') from your proofs.
  \item Always justify answers unless it is specifically stated not to do so.
  \item Cite theorems with proper references to chapters. For example, everyone's favorite, Theorem 4 from Chapter 1 can be written Theorem 1.4.

  \item The order in which you present information is important! If you are trying to prove an equality, do not start with the equality that you are trying to prove and show that it implies a true statement. For example,
        $$0=3 \implies 0=0,$$
        but that does not prove that $0=3$. Alternatively, you can start with one expression and show a chain of equalities that end with the other side of the expression you are trying to prove. For example, if you are trying to prove that $\frac{1}{(x+1)(x-2)}=\frac{-1}{2(x+1)}+\frac{1}{2(x-1)}$, you can show $$\frac{1}{(x+1)(x-1)}=\frac{\frac{x}{2}-\frac{x}{2}+\frac{1}{2}+\frac{1}{2}}{(x+1)(x-1)}=\frac{\frac{-1}{2}(x-1)+\frac{1}{2}(x+1)}{(x+1)(x-1)}=\frac{-1}{2(x+1)}+\frac{1}{2(x-1)},$$
        where each expression follows from the previous.
  \item If you are asked to find a solution to a matrix equation, $A\mathbf{x}={\bf b}$, present the solution in the form ${\bf x}=\begin{bmatrix}x_1\\ x_2 \\ \vdots \\ x_n\end{bmatrix}$. Do not just list the values of $x_1$, $x_2$, $\dots$, $x_n$, unless specifically directed.

\end{enumerate}

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\noindent NOTE: If you are interested in learning to type in \LaTeX (it is a good skill to have and comes in handy in many courses and situations), reach out to the department and they will put you in contact with me or another student who can help you learn it.


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% \item Doing row reductions horizontally is better organizationally, and remember to put the row operations above the similar symbol between matrices like this: $\overset{R_1\to R_1+R_2}{\sim}$.
% \item If a system has a free variable, this does not imply that the system has infinitely many solutions. If a column of a matrix is not a pivot column, it is not necessarily corresponding to a free variable. For example, the system corresponding to the matrix $$\begin{bmatrix}1 &0&0&0&0&0& 3\\0&1&0&0&0&0&4\end{bmatrix}$$ has no free variables.
% \item Do not confuse sets and matrices: $$\begin{bmatrix}a& b& c\end{bmatrix}\neq \{a,b,c\}.$$

% \item Do not start talking about a matrix for a transformation before you have noted that one even exists. Also not confuse the matrix for a transformation with the function itself. For example, do not say ``the transformation $T$ has $n$ pivots'' or ``$A$ maps from $\R^m$ to $\R^n$.''
% \item Sets of vectors or columns (or rows) of a matrix can be linearly (in)dependent, not matrices themselves.
% \item The matrix equation $A\vec{x}=0$ always has the trivial solution, $\vec{x}=0$.
% \item Nul $A$ refers to the nullspace of the matrix $A$, and so is a set. For example, the sets $\{0\}$, and $\text{span}\{(1,0,3),(1,2,0)\}$ could be the nullspace of a matrix. The nullity of $A$ is the dimension of the nullspace and so is a natural number.
% \item col $A$ is the columnspace of the matrix $A$ and so is a set. Writing $$\text{col } A = \{\text{columns of $A$}\}$$ is ridiculous. You may write, however, $$\text{col } A = \text{span}\{\text{columns of $A$}\}.$$
% \item col $A$ is only ever a finite set if col $A = \{0\}$. Otherwise, col $A$ is either a span of a set of vectors or is $\R^n$ for some $n\in\N$.
% \item Something can't `fail' the IMT. If the IMT says that $P\iff Q$, then $\text{\textlnot} P \iff \text{\textlnot} Q$ is also true. In another situation, you may be interested in a statement `contradicting' the IMT, if you are doing a proof by contradiction.
% \item The zero subspace is $\{0\}$, not 0. $$\text{Nul }A=0$$ doesn't make sense, but $$\text{dim Nul }A=\text{Nulltiy } A = 0$$ does make sense.


% \item Quantify variables and do so in the proper place. For example, ``for all  $b\in\R,\ 0\neq b$'' is not equivalent to ``$0\neq b$ for all $b\in\R$.''

% \item If transformation $T$ is linear, then  $T(\vec{0})=\vec{0}$. The converse of this statement, if $T(\vec{0})=\vec{0}$, then $T$ is linear, is false. That is, it is not true for all transformations $T$ that $T(\vec{0})=\vec{0}$ implies that $T$ is linear. There are plenty of non-linear transformations that send 0 to 0.

%\item Don't do this: ``Write $I_n=\begin{bmatrix}I_1&I_2&\dots&I_n\end{bmatrix}$.'' This would imply that $I_n$ is somehow recursively defined. You mean to write $$I_n=\begin{bmatrix}e_1&e_2&\dots&e_n\end{bmatrix}=\begin{bmatrix}1&0&0&\dots&0\\0&1&0&\dots&0\\0&0&1& & 0\\ \vdots &\vdots& & \ddots &\vdots\\ 0 & 0 & 0 & \dots & 1\end{bmatrix}.$$

% \item The word multiplied can take on many different meanings, so be clear in what sense you are using it. For example, to say that you ``multiply a row of $A$ and a column of $B$'' is vague.

% \item $\R^n=\{(x_1,x_2,\dots,x_n)\ : \ x_1,x_2,\dots,x_n\in\R\}$ is the set of all ordered tuples of $n$ real numbers, not $R^n$.

% \item When writing down an arbitrary matrix, do not change the letter of the entries. Instead, use two subscripts. For example, write, $$\begin{bmatrix}\vec{a}_{1}&\vec{a}_{2}&\dots&\vec{a}_{n}\end{bmatrix}=\begin{bmatrix}a_{1,1}&a_{2,1}&\dots&a_{n,1}\\ a_{1,2}&a_{2,2}&\dots&a_{n,2}\\ \vdots&\vdots& & \vdots\\ a_{1,m}&a_{2,m}&\dots&a_{n,m}\end{bmatrix}.$$

% \item \textit{Matrices} don't have solutions: \textit{systems} do.
% \item When writing the components of a \textit{vector}, they themselves are not also vectors, they are scalars, so do not have vector arrows above them. For example, $$\begin{bmatrix}\vec{a}&\vec{b}&\vec{c}\end{bmatrix}$$ is a matrix whose columns are the vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$, while $$\begin{bmatrix}a&b&c\end{bmatrix}$$ is a matrix whose three entries are scalars. Write $\vec{a}$ for a vector (include the arrow).

%\item Never draw $\dots$ where it is unclear what it means. For example, you can write $(1,0,\dots,0)$ for $(1,0,0,0,0,0,0,0,0)$, but not $(1,0,1,\dots,0)$ for $(1,0,1,0,0,0,0,0,0)$. Usually dots can be drawn when there is some arithmetic progression that is very obvious like $(1,2,3,4,\dots,104)$, or between entries that are identical like $(1,0,1,\dots,0)$ for $(1,0,1,0,0,0,0,0,0)$.

% \item Unnecessary notation is distracting. A common example of this is introducing variables such as $A$ and $\vec{b}$ or $\vec{x}$ to label matrices and vectors when it is not needed.
% \item If you are attempting to prove the equivalence of two expressions, do not start with the equality that you are trying to prove. Instead, begin with one side and write a chain of equalities that ends with the other side.

% \item A matrix-vector product is written $$\begin{bmatrix}a_1&a_2\\ a_3&a_4\end{bmatrix}\begin{bmatrix}x_1\\ x_2\end{bmatrix}=\begin{bmatrix}b_1\\ b_2\end{bmatrix} \ \ \ \ \ \  \text{not} \ \ \ \ \ \ \begin{bmatrix}a_1&a_2\\ a_3&a_4\end{bmatrix}\times\begin{bmatrix}x_1\\ x_2\end{bmatrix}=\begin{bmatrix}b_1\\ b_2\end{bmatrix}.$$

% \item The word `assume' is not used in place of `let' or `write'. If you are proving something about an arbitrary vector $\vec{u}\in\R^n$, you may say,

% $$\text{let } \vec{u}=\begin{bmatrix}u_1\\\vdots\\ u_n\end{bmatrix}\ \ \ \ \ \ \text{or}\ \ \ \ \ \ \text{write } \vec{u}=\begin{bmatrix}u_1\\\vdots\\ u_n\end{bmatrix}.$$

% \item It is \textbf{not} generally true that for all $a,b\in\R$ and all $\vec{x},\vec{y}\in\R^n$, $$a\vec{x}+b\vec{y}=0 \implies a=0 \text{ and }b=0.$$

% \item It is possible that a linearly dependent set of vectors in $\R^n$ spans $\R^n$.

% \item If it is possible to prove or explain something without using a contradiction technique, it is better to avoid doing so. That being said, if you are trying to \textit{disprove} something, it is usually simplest to provide a counterexample.

% \item If you are asked to prove $P\implies Q$, do not prove that $Q\implies P$.