Tables, Lists and Matrices. MAT 218 Mathematical Computing. February 2, 2016

Transcription

1 Tables, Lists and Matrices MAT 218 Mathematical Computing February 2, List Environments Before we get started, I want to point out two things I have in the preamble The first is that I am using the package \usepackage{fullpage}, which makes wider margins and makes better use of the space on a page, so I usually use it The other item from the preamble is to make no numbers on the bottom of the pages To make a document without page numbers, we use the \pagestyle{empty} command 11 Enumerate Often we want to list items and number them along the way To do this, we use the enumerate command In order to do so, we need to begin and end the section Each element is called an item The format for this is : \begin{enumerate} \item First thing \item Second thing \item Last thing \end{enumerate} So, suppose we wanted to list the NFL championship weekend playoff teams Using the notation above, those teams are : 1 New England Patriots 2 Denver Broncos 3 Carolina Panthers 4 Arizona Cardinals

2 12 Itemize We sometimes do not want numbered lists, however Sometimes we just want the list to have bullets To do this, we use the itemize command We do not get a choice of what the bullet will look like, though But the format is exactly the same as it is for the enumerate command So, suppose we wanted to list the last 4 teams projected to make the NCAA tournament These teams are not ranked, so numbering them wouldn t make a lot of sense If we use the itemize command, we would get a list that looks like : Notre Dame Connecticut Rhode Island San Diego State It is important to note that regardless of which environment we are defining, you must begin and end the environment as described above If you forget to end the section, you will get errors, as you will if you use the wrong kind of brackets 13 Description We can also make lists where we have no symbols or numbers in front of the items To do this, we use the description environment, which uses the exact same format as the other environments The following is the code and then the output for the description section \centerline{ways to Emphasize Text} \begin{description} \item \textbf{boldface} \item \textit{italicize} \item \underline{underline} \end{description} Ways to Emphasize Text boldface italicize underline

3 2 Quotes The quote environment works in a similar way to the list environments in that we have to begin and end the section with \begin{quote} and \end{quote} but differs in that we do not have items to list out Once we begin the section, whatever we write inside those two commands will be centered and indented - just like the quotes you see in books If we want to change the size of the font, we can put a command inside the environment and the size will automatically revert back to the normal size when we close the environment Here, I am beginning the section with \begin{quote} \small % a smaller font will be used inside this environment and then after typing what I want, the section ends with \end{quote} So suppose we wanted to include words of wisdom from Denis Leary In this environment, the quote would look like this: Racism isn t born, folks, it s taught I have a two-year-old son You know what he hates? Naps! End of list 3 Tables Suppose we wanted to list the 2015 Red Sox starting lineup along with the number of homeruns and RBIs each player had We would use the tabular environment to do this The format is similar to that of before, with a beginning and end, but we also need to include additional information, as in the location of the values within the cells and whether or not we want consistent lines in the table The format for the end is simply \end{tabular}, but the start is of the form \begin{tabular}{conditions} These conditions are either l,c or r depending on whether we want the item to be left justified, centered or right justified in the cell Also, the symbols in the conditions section are used to put a vertical line in each row And, if we want a horizontal line, we include the \hline command at the end of the line - one for each line we want If we wanted vertical bars separating the columns, and horizontal lines separating the rows, the syntax would look like : \begin{center} \begin{tabular}{ l c c } \hline \textbf{name} & \textbf{homeruns} & \textbf{rbis}\\ \hline \hline Blake Swihart & 5 & 31\\ \hline Mike Napoli & 13 & 40\\ \hline Dustin Pedroia & 12 & 42\\ \hline Xander Bogaerts & 7 & 81\\ \hline Pablo Sandoval & 10 & 47\\ \hline

4 Hanley Ramirez & 19 & 53\\ \hline Mookie Betts & 18 & 77\\ \hline Rusney Castillo & 5 & 29\\ \hline David Ortiz & 37 & 108\\ \hline\hline \end{tabular} \end{center} Note: We use & between items that are to be in different cells We also need to put a manual carriage return in at the end of each line This is done with the \\ command The output associated with the above code would be: Name Homeruns RBIs Blake Swihart 5 31 Mike Napoli Dustin Pedroia Xander Bogaerts 7 81 Pablo Sandoval Hanley Ramirez Mookie Betts Rusney Castillo 5 29 David Ortiz But we may want to have some of the lines follow a different format by using the multicolumn command This is used when we want to define a line differently than we did the rows in the beginning of the environment The syntax for the command is \multicolumn{# of columns}{conditions}{any text} So, if we wanted the above table to have a title at the top that had the team name over the first column but had the word Statistics centered over the other two, we could put this command as the first line in the table: \multicolumn{1}{ l }{2015 Red Sox} & \multicolumn{2}{c }{Statistics}\\ \hline The important thing to be sure of here is that the sum of the number of columns in this command must match the number of columns in the regular table assignment The new table would look like :

5 2015 Red Sox Statistics Name Homeruns RBIs Blake Swihart 5 31 Mike Napoli Dustin Pedroia Xander Bogaerts 7 81 Pablo Sandoval Hanley Ramirez Mookie Betts Rusney Castillo 5 29 David Ortiz We can combine two different types of environments as well For example we can combine the itemize and tabular environment to do things like : The 2015 Red Sox Starting Lineup Mooke Betts Dustin Pedroia Xander Boegaerts Blake Swihart CF 2B SS C The 2016 Projected Red Sox Starting Lineup Mooke Betts Dustin Pedroia Xander Bogaerts Jackie Bradley Jr 4 Matrices CF 2B SS RF Now, we will look at how to make a variety of matrices We will also look at how we can reference equations in text - as in if we labeled a matrix or an equation in a paper and then wanted to refer to it - L A TEXhas a way to code this We can use the array environment to make matrices, as well as to display other mathematical structures The standard L A TEXsyntax for an array is \begin{array}[pos]{preamble} row-list \end{array} The optional arguments [pos] can be center, c, top t, or bottom, b The purpose of this to specify how each row is aligned, but this command is unnecessary The {preamble} determines whether columns are centered, c, left-justified, l or right-justified, r

6 So, suppose we wanted to make an 3 3 array with the letters a i,j as the entries We can use this environment to do so relatively easily \begin{array}{ccc} a_{1,1} & a_{1,2} & a_{1,3}\\ a_{2,1} & a_{2,2} & a_{2,3}\\ a_{3,1} & a_{3,2} & a_{3,3} \end{array} a 1,1 a 1,2 a 1,3 a 2,1 a 2,2 a 2,3 a 3,1 a 3,2 a 3,3 In order to put some kind of border around an array you must use the \left and \right commands \[A=\left[\begin{array}{ccc} a_{11}&a_{12}& a_{13}\\ a_{21}&a_{22}& a_{23}\\ a_{31}&a_{32}& a_{33} \end{array}\right]\] A = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 There are other types of borders we could use, like ( ) { a11 a A = 12 1, x < 0, f(x) = a 21 a 22 1, x 0 We can also easily add horizontal or vertical bars within the matrix as long as we want the bar to span the entire matrix For vertical bars, we insert in the appropriate place in the preamble If we want a horizontal line, we use the \hline command \[A=\left[\begin{array}{cc c} a_{11}&a_{12}& a_{13}\\ a_{21}&a_{22}& a_{23}\\ \hline a_{31}&a_{32}& a_{33} \end{array}\right]\] A = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 But, what if we didn t want the horizontal line to span the entire matrix? Then you need to use \cline{n-m} which puts a horizontal line from column n to m In this example I have also introduced the use of multicolumn to just put a vertical bar between two columns in one row To make this easier, I also defined the command to produce this vertical line Now, any time I use \tempb, it will produce that verical line in a matrix The syntax for the multicolumn is to put the number of columns, then the location (with the vertical bars as desired), and then then the text that you want to be in those cells \[ \newcommand*{\tempb}{\multicolumn{1}{ c}{b}} P=\left[ \begin{array}{ccc} A & A & \tempb \\ \cline{1-2} C & C & D \end{array}\right] \] P = [ A A B C C D ]

7 Here are some other things we can do \renewcommand{\arraystretch}{2} $$\newcommand*{\temp}{\multicolumn{1}{r }{}} A=\left[\begin{array}{cccccc} 1 & 2 &3 & \temp & 7 & 6\\ \cline{1-6} 2 & 4 & 6 &\temp & 5 & 4\\ \end{array}\right] $$ A = $$\newcommand*{\temp}{\multicolumn{1}{c }{0}} B=\left[\begin{array}{cccccc} 1 & 0 & \ast & 0 & \ast &\ast \\ \cline{1-1} \temp & 1 & \ast & 0 &\ast & \ast \\ \cline{2-3} 0 & 0 & \temp & 1 & \ast & \ast \\ \cline{4-6} 0 & 0 & 0& 0 & 0 & 0 \end{array}\right] $$ B = One final tool for doing this kind of block structure is the hhline package which must be requested in the preamble with \usepackage{hhline} \setlength{\arrayrulewidth}{5pt} \[ G=\left[ \begin{array}{ cc c c } \hhline{ t:==:t:==:t } a&b&c&d\\ \hhline{ :==: ~ ~ } 1&2&3&4\\ \hhline{#==#~ =#} i&j&k&l\\ \hhline{ } w&x&y&z\\ \hhline{ b:==:b:==:b } \end{array}\right] \] G = a b c d i j k l w x y z It takes a little playing with the parameters to get a good feel of how they work, so don t get discouraged if your array doesn t look the way you want right away This environment is not always the best to use when making matrices It can be easier for us to use matrix environment when we want to make basic matrices In order to use this, you must open the AMSMATH package in the preamble of the document Once we do, we can use the following matrix environments: 1 \begin{matrix} 1&2\\3&4 \end{matrix}:

8 2 \begin{bmatrix} 1&2\\3&4 \end{bmatrix}: 3 \begin{pmatrix} 1&2\\3&4 \end{pmatrix}: 4 \begin{vmatrix} 1&2\\3&4 \end{vmatrix}: 5 \begin{vmatrix} 1&2\\3&4 \end{vmatrix}: [ ] ( ) Note that any mathematical expressions can be the entries 1 0 ex cos(x) dx (x 2 d + 1) dx sin2 (x) a x

9 5 Your Assignment 1 Make an ordered list of the top 5 CDs you think everyone should own 2 Make a bulleted list of your 5 favorite movies For each movie, make a bulleted sublist under the name of the movie with 3 actors or actresses who starred in the movie 3 Make a table with each of the math classes you have taken in one column, the grade you received in the a second column and the semester you took the class in a third Using the multicolumn command, put in a line at the top of the table that serves as a title (ie one cell instead of 3) and use the multicolumn command at the end to make a definition for the last line where Grade Point Average is left justified over the first 2 columns and the GPA you have earned in your math classes is given in the center of the 3 rd column 4 Reproduce exactly the following matrix 1 α 1 α1 2 α n α 2 α2 2 α n α 3 α3 2 α3 n 1 1 α m αm 2 αm n 1 the dots commands are \vdots, \cdots and ddots Play with them to see what they do 5 Build the matrix [ A B F = C D ] (1) This assignment is due by the start of class on Tuesday, February 9 th