Lists

Making Lists of Objects	Adding, Removing, and Modifying List Elements
Collecting Objects Together	Combining Lists
Making Tables of Values	Lists as Sets
Manipulating Elements of Lists	Rearranging Lists
Vectors and Matrices	Grouping and Combining Elements of Lists
Getting Pieces of Lists	Ordering in Lists
Testing and Searching List Elements	Rearranging Nested Lists

Making Lists of Objects

In doing calculations, it is often convenient to collect together several objects, and treat them as a single entity. Lists give you a way to make collections of objects in the Wolfram Language. As you will see later, lists are very important and general structures in the Wolfram Language.

A list such as {3,5,1} is a collection of three objects. But in many ways, you can treat the whole list as a single object. You can, for example, do arithmetic on the whole list at once, or assign the whole list to be the value of a variable.

Here is a list of three numbers:

This squares each number in the list, and adds 1 to it:

This takes differences between corresponding elements in the two lists. The lists must be the same length:

The value of % is the whole list:

You can apply any of the mathematical functions in "Some Mathematical Functions" to whole lists:

Just as you can set variables to be numbers, so also you can set them to be lists.

This assigns v to be a list:

Wherever v appears, it is replaced by the list:

Collecting Objects Together

We first encountered lists in "Making Lists of Objects" as a way of collecting numbers together. Here, we shall see many different ways to use lists. You will find that lists are some of the most flexible and powerful objects in the Wolfram Language. You will see that lists in the Wolfram Language represent generalizations of several standard concepts in mathematics and computer science.

At a basic level, what a Wolfram Language list essentially does is to provide a way for you to collect together several expressions of any kind.

Here is a list of numbers:

This gives a list of symbolic expressions:

You can differentiate these expressions:

And then you can find values when x is replaced with 3:

The mathematical functions that are built into the Wolfram Language are mostly set up to be "listable" so that they act separately on each element of a list. This is, however, not true of all functions in the Wolfram Language. Unless you set it up specially, a new function f that you introduce will treat lists just as single objects. "Applying Functions to Parts of Expressions" and "Structural Operations" will describe how you can use Map and Thread to apply a function like this separately to each element in a list.

Making Tables of Values

You can use lists as tables of values. You can generate the tables, for example, by evaluating an expression for a sequence of different parameter values.

This gives a table of the values of

, with

running from 1 to 6:

Here is a table of

for

from

This gives the numerical values:

You can also make tables of formulas:

Table uses exactly the same iterator notation as the functions Sum and Product, which are discussed in "Sums and Products":

This makes a table with values of x running from 0 to 1 in steps of 0.25:

You can perform other operations on the lists you get from Table:

TableForm displays lists in a "tabular" format. Notice that both words in the name TableForm begin with capital letters:

All the examples so far have been of tables obtained by varying a single parameter. You can also make tables that involve several parameters. These multidimensional tables are specified using the standard Wolfram Language iterator notation, discussed in "Sums and Products".

This makes a table of

with

running from 1 to 3 and

running from 1 to 2:

The table in this example is a list of lists. The elements of the outer list correspond to successive values of

. The elements of each inner list correspond to successive values of

, with

fixed.

Sometimes you may want to generate a table by evaluating a particular expression many times, without incrementing any variables.

This creates a list containing four copies of the symbol x:

This gives a list of four pairs of numbers sampled from {1,2,3,4}. Table reevaluates RandomSample[{1,2,3,4},2] for each element in the list, so that you get four different samples:

This evaluates

for each of the values of i in the list {1,4,9,16}:

This creates a 3×2 table:

In this table, the length of the rows depends on the more slowly varying iterator variable, i:

You can use Table to generate arrays with any number of dimensions.

This generates a three‐dimensional 2×2×2 array. It is a list of lists of lists:

Table[f,{i_max}]	give a list of i_max values of f
Table[f,{i,i_max}]	give a list of the values of f as i runs from 1 to i_max
Table[f,{i,i_min,i_max}]	give a list of values with i running from i_min to i_max
Table[f,{i,i_min,i_max,di}]	use steps of di
Table[f,{i,i_min,i_max},{j,j_min,j_max},…]	generate a multidimensional table
Table[f,{i,{i₁,i₂,…}]	give a list of the values of f as i successively takes the values i₁, i₂, …
TableForm[list]	display a list in tabular form

Functions for generating tables.

You can use the operations discussed in "Manipulating Elements of Lists" to extract elements of the table.

This creates a table and gives it the name sq:

This gives the third part of the table:

This gives a list of the third through fifth parts:

This creates a 2×2 table, and gives it the name m:

This extracts the first sublist from the list of lists that makes up the table:

This extracts the second element of that sublist:

This does the two operations together:

This displays m in a "tabular" form:

t[[i]] or Part[t,i]	give the i ^th sublist in t (also input as t〚i〛)
t[[i;;j]] or Part[t,i;;j]	give a list of the parts i through j
t[[{i₁,i₂,…}]] or Part[t,{i₁,i₂,…}]	give a list of the i₁ ^th, i₂ ^th, … parts of t
t[[i,j,…]] or Part[t,i,j,…]	give the part of t corresponding to t[[i]][[j]]…

Ways to extract parts of tables.

As mentioned in "Manipulating Elements of Lists", you can think of lists in the Wolfram Language as being analogous to "arrays". Lists of lists are then like two‐dimensional arrays. When you lay them out in a tabular form, the two indices of each element are like its

and

coordinates.

Manipulating Elements of Lists

Many of the most powerful list manipulation operations in the Wolfram Language treat whole lists as single objects. Sometimes, however, you need to pick out or set individual elements in a list.

You can refer to an element of a Wolfram Language list by giving its "index". The elements are numbered in order, starting at 1.

{a,b,c}	a list
Part[list,i] or list[[i]]	the i ^th element of list (the first element is list[[1]])
Part[list,{i,j,…}] or list[[{i,j,…}]]	a list of the i ^th, j ^th, … elements of list
Part[list,i;;j]	a list of the i ^th through j ^th elements of list

Operations on list elements.

This extracts the second element of the list:

This extracts a list of elements:

This assigns the value of v to be a list:

You can extract elements of v:

By assigning a variable to be a list, you can use Wolfram Language lists much like "arrays" in other computer languages. Thus, for example, you can reset an element of a list by assigning a value to v[[i]].

Part[v,i] or v[[i]]	extract the i ^th element of a list
Part[v,i]=value or v[[i]]=value	reset the i ^th element of a list

Array‐like operations on lists.

Here is a list:

This resets the third element of the list:

Now the list assigned to v has been modified:

Vectors and Matrices

Vectors and matrices in the Wolfram Language are simply represented by lists and by lists of lists, respectively.

{a,b,c}	vector
{{a,b},{c,d}}	matrix

The representation of vectors and matrices by lists.

This is a 2×2 matrix:

Here is the first row:

Here is the element

This is a two‐component vector:

The objects p and q are treated as scalars:

Vectors are added component by component:

This gives the dot (scalar) product of two vectors:

You can also multiply a matrix by a vector:

Or a matrix by a matrix:

Or a vector by a matrix:

This combination makes a scalar:

Because of the way the Wolfram Language uses lists to represent vectors and matrices, you never have to distinguish between "row" and "column" vectors.

Table[f,{i,n}]	build a length‐n vector by evaluating f with i=1,2,…,n
Array[a,n]	build a length‐n vector of the form {a[1],a[2],…}
Range[n]	create the list {1,2,3,…,n}
Range[n₁,n₂]	create the list {n₁,n₁+1,…,n₂}
Range[n₁,n₂,dn]	create the list {n₁,n₁+dn,…,n₂}
list[[i]] or Part[list,i]	give the i ^th element in the vector list
Length[list]	give the number of elements in list
c v	multiply a vector by a scalar
a.b	dot product of two vectors
Cross[a,b]	cross product of two vectors (also input as a×b )
Norm[v]	Euclidean norm of a vector

Functions for vectors.

Table[f,{i,m},{j,n}]	build an m×n matrix by evaluating f with i ranging from 1 to m and j ranging from 1 to n
Array[a,{m,n}]	build an m×n matrix with i,j ^th element a[i,j]
IdentityMatrix[n]	generate an n×n identity matrix
DiagonalMatrix[list]	generate a square matrix with the elements in list on the main diagonal
list[[i]] or Part[list,i]	give the i ^th row in the matrix list
list[[All,j]] or Part[list,All,j]	give the j ^th column in the matrix list
list[[i,j]] or Part[list,i,j]	give the i,j ^th element in the matrix list
Dimensions[list]	give the dimensions of a matrix represented by list

Functions for matrices.

Column[list]	display the elements of list in a column
MatrixForm[list]	display list in matrix form

Formatting constructs for vectors and matrices.

This builds a 3×3 matrix

with elements

This displays s in standard two‐dimensional matrix format:

This gives a vector with symbolic elements. You can use this in deriving general formulas that are valid with any choice of vector components:

This gives a 3×2 matrix with symbolic elements. "Building Lists from Functions" discusses how you can produce other kinds of elements with Array:

Here are the dimensions of the matrix on the previous line:

This generates a 3×3 diagonal matrix:

c m	multiply a matrix by a scalar
a.b	dot product of two matrices
Inverse[m]	matrix inverse
MatrixPower[m,n]	n ^th power of a matrix
Det[m]	determinant
Tr[m]	trace
Transpose[m]	transpose
Eigenvalues[m]	eigenvalues
Eigenvectors[m]	eigenvectors

Some mathematical operations on matrices.

Here is the 2×2 matrix of symbolic variables that was defined:

This gives its determinant:

Here is the transpose of m:

This gives the inverse of m in symbolic form:

Here is a 3×3 rational matrix:

This gives its inverse:

Taking the dot product of the inverse with the original matrix gives the identity matrix:

Here is a 3×3 matrix:

Eigenvalues gives the eigenvalues of the matrix:

This gives a numerical approximation to the matrix:

Here are numerical approximations to the eigenvalues:

"Linear Algebra in Wolfram Language" discusses many other matrix operations that are built into the Wolfram Language.

Getting Pieces of Lists

First[list]	the first element in list
Last[list]	the last element
Part[list,n] or list[[n]]	the n ^th element
Part[list,-n] or list[[-n]]	the n ^th element from the end
Part[list,m;;n]	elements m through n
Part[list,{n₁,n₂,…}] or list[[{n₁,n₂,…}]]
	the list of elements at positions n₁, n₂, …

Picking out elements of lists.

We will use this list for the examples:

Here is the last element of t:

This gives the third element:

This gives the list of elements 3 through 6:

This gives a list of the first and fourth elements:

Take[list,n]	the first n elements in list
Take[list,-n]	the last n elements
Take[list,{m,n}]	elements m through n (inclusive)
Rest[list]	list with its first element dropped
Drop[list,n]	list with its first n elements dropped
Most[list]	list with its last element dropped
Drop[list,-n]	list with its last n elements dropped
Drop[list,{m,n}]	list with elements m through n dropped

Picking out sequences in lists.

This gives the first three elements of the list t defined above:

This gives the last three elements:

This gives elements 2 through 5 inclusive:

This gives elements 3 through 7 in steps of 2:

This gives t with the first element dropped:

This gives t with its first three elements dropped:

This gives t with only its third element dropped:

"Manipulating Expressions like Lists" shows how all the functions here can be generalized to work not only on lists, but on any Wolfram Language expressions.

The functions here allow you to pick out pieces that occur at particular positions in lists. "Finding Expressions That Match a Pattern" shows how you can use functions like Select and Cases to pick out elements of lists based not on their positions, but instead on their properties.

Testing and Searching List Elements

Position[list,form]	the positions at which form occurs in list
Count[list,form]	the number of times form appears as an element of list
MemberQ[list,form]	test whether form is an element of list
FreeQ[list,form]	test whether form occurs nowhere in list

Testing and searching for elements of lists.

"Getting Pieces of Lists" discusses how to extract pieces of lists based on their positions or indices. The Wolfram System also has functions that search and test for elements of lists, based on the values of those elements.

This gives a list of the positions at which a appears in the list:

Count counts the number of occurrences of a:

This shows that a is an element of {a,b,c}:

On the other hand, d is not:

This assigns m to be the 3×3 identity matrix:

This shows that 0 does occur somewhere in m:

This gives a list of the positions at which 0 occurs in m:

As discussed in "Finding Expressions That Match a Pattern", the functions Count and Position, as well as MemberQ and FreeQ, can be used not only to search for particular list elements, but also to search for classes of elements which match specific "patterns".

Adding, Removing, and Modifying List Elements

Prepend[list,element]	add element at the beginning of list
Append[list,element]	add element at the end of list
Insert[list,element,i]	insert element at position i in list
Insert[list,element,-i]	insert at position i counting from the end of list
Riffle[list,element]	interleave element between the entries of list
Delete[list,i]	delete the element at position i in list
ReplacePart[list,i->new]	replace the element at position i in list with new
ReplacePart[list,{i,j}->new]	replace list[[i,j]] with new

Functions for manipulating elements in explicit lists.

This gives a list with x prepended:

This inserts x so that it becomes element number 2:

This interleaves x between the entries of the list:

This replaces the third element in the list with x:

This replaces the 1, 2 element in a 2×2 matrix:

Functions like ReplacePart take explicit lists and give you new lists. Sometimes, however, you may want to modify a list "in place", without explicitly generating a new list.

v={e₁,e₂,…}	assign a variable to be a list
v[[i]]=new	assign a new value to the i ^th element

Resetting list elements.

This defines v to be a list:

This sets the third element to be x:

Now v has been changed:

m[[i,j]]=new	replace the (i,j) ^th element of a matrix
m[[i]]=new	replace the i ^th row
m[[All,i]]=new	replace the i ^th column

Resetting pieces of matrices.

This defines m to be a matrix:

This sets the first column of the matrix:

This sets every element in the first column to be 0:

Combining Lists

Join[list₁,list₂,…]	concatenate lists together
Union[list₁,list₂,…]	combine lists, removing repeated elements and sorting the result
Riffle[list₁,list₂]	interleave elements of list₁ and list₂

Functions for combining lists.

Join concatenates any number of lists together:

Union combines lists, keeping only distinct elements:

Riffle combines lists by interleaving their elements:

Lists as Sets

The Wolfram Language usually keeps the elements of a list in exactly the order you originally entered them. If you want to treat a Wolfram Language list like a mathematical set, however, you may want to ignore the order of elements in the list.

Union[list₁,list₂,…]	give a list of the distinct elements in the list_i
Intersection[list₁,list₂,…]	give a list of the elements that are common to all the list_i
Complement[universal,list₁,…]	give a list of the elements that are in universal, but not in any of the list_i
Subsets[list]	give a list of all subsets of the elements in list
DeleteDuplicates[list]	delete all duplicates from list

Set theoretical functions.

Union gives the elements that occur in any of the lists:

Intersection gives only elements that occur in all the lists:

Complement gives elements that occur in the first list, but not in any of the others:

This gives all the subsets of the list:

DeleteDuplicates deletes all duplicate elements from the list:

Rearranging Lists

Sort[list]	sort the elements of list into a standard order
Union[list]	sort elements, removing any duplicates
Reverse[list]	reverse the order of elements in list
RotateLeft[list,n]	rotate the elements of list n places to the left
RotateRight[list,n]	rotate n places to the right

Functions for rearranging lists.

This sorts the elements of a list into a standard order. In simple cases like this, the order is alphabetical or numerical:

This sorts the elements, removing any duplicates:

This rotates ("shifts") the elements in the list two places to the left:

You can rotate to the right by giving a negative displacement, or by using RotateRight:

PadLeft[list,len,x]	pad list on the left with x to make it length len
PadRight[list,len,x]	pad list on the right

Padding lists.

This pads a list with x's to make it length 10:

Grouping and Combining Elements of Lists

Partition[list,n]	partition list into n‐element pieces
Partition[list,n,d]	use offset d for successive pieces
Split[list]	split list into pieces consisting of runs of identical elements
SplitBy[list,f]	split list into runs of elements with identical values when f is applied
Gather[list]	gather the elements of list into sublists of identical elements
GatherBy[list,f]	gather the elements of list into sublists with identical values when f is applied

Functions for grouping together elements of lists.

Here is a list:

This groups the elements of the list in pairs, throwing away the single element left at the end:

This groups elements in triples. There is no overlap between the triples:

This makes triples of elements, with each successive triple offset by just one element:

This splits up the list into runs of identical elements:

Here is a random list of integers:

This splits the list into runs of even and odd numbers:

Gather the integers into sublists:

Gather the integers into sublists based on equivalence mod 3:

Tuples[list,n]	generate all possible n‐tuples of elements from list
Tuples[{list₁,list₂,…}]	generate all tuples whose i ^th element is from list_i

Finding possible tuples of elements in lists.

This gives all possible ways of picking two elements out of the list:

This gives all possible ways of picking one element from each list:

Ordering in Lists

Sort[list]	sort the elements of list into order
Ordering[list]	the positions in list of the elements in Sort[list]
Ordering[list,n]	the first n elements of Ordering[list]
Ordering[list,-n]	the last n elements of Ordering[list]
Permutations[list]	all possible orderings of list
Min[list]	the smallest element in list
Max[list]	the largest element in list

Ordering in lists.

Here is a list of numbers:

This gives the elements of t in sorted order:

This gives the positions of the elements of t, from the position of the smallest to that of the largest:

This is the same as Sort[t]:

This gives the smallest element in the list:

Rearranging Nested Lists

You will encounter nested lists if you use matrices or generate multidimensional arrays and tables. The Wolfram Language provides many functions for handling such lists.

Flatten[list]	flatten out all levels in list
Flatten[list,n]	flatten out the top n levels in list
Partition[list,{n₁,n₂,…}]	partition into blocks of size n₁×n₂×…
Transpose[list]	interchange the top two levels of lists
RotateLeft[list,{n₁,n₂,…}]	rotate successive levels by n_i places
PadLeft[list,{n₁,n₂,…}]	pad successive levels to be length n_i

A few functions for rearranging nested lists.

This "flattens out" sublists. You can think of it as effectively just removing all inner braces:

This flattens out only one level of sublists:

There are many other operations you can perform on nested lists. More operations are discussed in "Manipulating Lists".

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