Educational Materials

Introduction to Mathematica for High School Math (for Students and Teachers) #4

Ruth Dover

Author

Ruth Dover

Title

Introduction to Mathematica for High School Math (for Students and Teachers) #4

Description

Introduction to Mathematica for High School Math

Tutorial 4: Lists and Tables

R. Dover, IMSA

Lists and Tables

Curly braces are used a lot in Mathematica®. They are used primarily for lists of objects, but they are also used for indices and for domains of functions.

First, we will create a short list and name it. Then we will perform a few simple operations on it:

mylist={2,3,5,8,13}

mylist

mylist+1

Clearly, the need may come to generate a list. Here, this is called a

Table

mytable=Table[3i,{i,1,5}]

Note that the result is in fact a list, since the output is in curly braces. Following is another type of command that is frequently used with lists. Try some other numbers besides 4 and be sure you understand what it does:

mytable[[4]]

By default, the step size for a Table is 1, though this may be controlled by an optional value within the iterator. Also, we make a table of two values in the following:

Table[{x,Cos[x]},{x,0,2π,π/4}]

This is rather difficult to view. Use the command

TableForm

on the last command. (Either type

//TableForm

after the command or put

TableForm[]

around the entire command. Notice the capital letters on both words, with no space between them, since it is one Mathematica command.)

Of course, there are more possibilities on the palette: under Basic Commands, the fifth button is List. There is a button for a basic Table, some commands for lists, and there is a drop-down button for more possibilities.

Define a function

of your choice. Then create a table that will show ordered pairs

(x,f(x))

for

0≤x≤5

by steps of 0.2, and display it with TableForm.

When you are done, be sure to clear

mylist

and

mytable

More Numerics

Did you notice that the terms in the last table you created came out in decimal form? Curious. Previously, we had to tell Mathematica to give answers in decimal form. Here, note that you entered decimals for the step size. Decimals in, decimals out—just as your calculator does. Try this on a couple of other commands.

Another Option

Here is another way to create a list—one that is often quite useful:

Range[6]

Range[1,10,2]

Displaying Lists: A Brief Intro

littlelist={1,2,4,4,5}

ListPlot[littlelist]

ListLinePlot[littlelist]

PieChart[littlelist]

BarChart[littlelist]

And as you can guess, all of these may be controlled in lots of ways: colors, labels, etc.

Vectors

Now we are going to create and name two vectors,

and

. (The letters are not capitalized to distinguish them from built-in names.) Vectors are simply lists:

v={2,-5,1};w={1,2,-3}

Find the vector

. Then find

2v−3w

Execute the next cell. What does the program assume about

c+v

The dot product is denoted by

v.w

, with a period between the two vectors. Try it. Then ask Mathematica if this operation on these vectors is commutative.

Try the cross product. As usual, this may be typed; or in the palette, under Basic Commands, choose the fifth button for List that you looked at previously. At the bottom right, find Vectors, with an assortment of commands:

Cross[v,w]

Alternatively, you may use a cute little "" created by Esc cross Esc. (That is not multiplication!):

vw

When finished, clear

and

Matrices

Matrices may be expressed in two ways. Either way, Mathematica sees a matrix as a list of lists. First we will define

mat1

by using the palette:

mat1=

3	5
-1	2



Curious output. This looks like the list of lists mentioned earlier with braces within braces. Select the cell by clicking the bracket on the right. Under the Cell menu, select Convert to and choose TraditionalForm. This will look nicer, but Mathematica is sometimes less confident of this notation. Note the new markings on the cell bracket. Alternatively, create

mat2

as follows and, on another line, use the command

MatrixForm[mat2]

mat2={{2,0},{3,1}}

Several commands work as expected on matrices. Find 3mat1: find the determinant by using

Det[mat1]

, and then find

MatrixPower[mat1,3]

, the third power of the matrix. Express these using

MatrixForm

. Also, you may type

//MatrixForm

after any expression.

Then find the

Transpose

and the

Inverse

of mat2.

Now find mat1+mat2 and mat1.mat2, using the period to represent multiplication as we did before. What happens if one enters

mat1

? Try it.

Check this out. What should the result be?

a	b
c	d

.Inverse

a	b
c	d



"Fix" that by using

Simplify

on the preceding command. Look better?

Whatever Size You Want

Back to the palette. Under Basic Commands, choose the fourth button, the one that looks like a little matrix! Many buttons should be self-explanatory. Define mat3, and then click the basic 2×2 matrix. Then click Add Row or Add Column as you wish. Fill in the boxes.

Now determine what each of the following commands does. (Hopefully your matrix is not too small!):

mat3[[2]]

mat3[[3,2]]

Exercises

Define

f(x)=sin(x)

and

g(x)=-.042

+1.31x-0.05

. Create a table of values to show

f(x)

and

g(x)

on the interval

−1≤x≤4

with a step size of 0.25. Use TableForm to display the results. What do you notice?

Find the dot product of the vectors

〈6,–8〉

and

〈4,3〉

. (What does the result tell you about the two vectors?)

Name and create a 4×2 matrix. Enter your own values. Then name and create a 2×3 matrix. Multiply the two matrices and display the result as a matrix.

Cite this as: Ruth Dover, "Introduction to Mathematica for High School Math (for Students and Teachers) #4" from the Notebook Archive (2021), https://notebookarchive.org/2021-09-6h1zkpf