# List

{e_{1},e_{2},…}

is a list of elements.

# Details

- Lists are very general objects that represent collections of expressions.
- Functions with attribute Listable are automatically "threaded" over lists, so that they act separately on each list element. Most built‐in mathematical functions are Listable.
- {a,b,c} represents a vector.
- {{a,b},{c,d}} represents a matrix.
- Nested lists can be used to represent tensors.
- If Nothing appears in a list, it is automatically removed.

# Background & Context

- List is a very general construct used to represent collections of expressions. Lists may have any length or depth. The expression List[a,b,c,…] is commonly written and displayed using the shorthand syntax {a,b,c,…}. Lists are particularly important in the Wolfram Language, which does not define explicit vector, matrix, tensor, etc. objects but rather uses (possibly nested) lists to represent such structures. For example, {a,b,c,…} can represent a vector, {{a,b},{c,d}} a matrix, and so on.
- Functions with attribute Listable are automatically “threaded” over lists, meaning they act separately on each list element. Most built‐in mathematical functions are Listable.
- Apply replaces the head of a List (or any other expression) with a new head, while Map applies a function to elements on the first level of a List (or any other expression).
- SparseArray may be used to efficiently represent and compute with lists (or nested lists) that have a constant (often 0) “background” value. A SparseArray can be expanded to a full-dimensional List using Normal.
- Values of a list can be efficiently modified in place using Set, e.g. list[[k]]=newValue. Common operations to access, insert, or delete elements of a list include Part, Take, Drop, Extract, Insert, Delete, PadLeft/PadRight, Append/AppendTo, and Prepend/PrependTo.
- A flat list of values (i.e. a vector) may be plotted using ListPlot, and an array of values given by a rectangular list of lists may be plotted using ArrayPlot, MatrixPlot, ListDensityPlot, or related functions. Other important and useful functions commonly applied to lists include Total, Accumulate, Mean, and ListConvolve.
- Association provides a generalization of symbolically indexed lists, associative arrays, dictionaries, hashmaps, structs, and a variety of other powerful data structures. An Association is so named because it associates keys with values, allowing highly efficient lookup and updating even with millions of elements.
- A list can be converted to a sequence of expressions by applying Sequence to it. This can be particularly useful since functions in the Wolfram Language often take a flat sequence of arguments instead of an argument list, so use of Sequence allows list-represented data to be easily spliced into other functions.

# Examples

open allclose all## Basic Examples (1)

## Scope (26)

### Representation of Vectors, Matrices, and Other Arrays (4)

A vector is a list of nonlist elements:

Many operations work on vectors, like Dot and Norm:

A matrix is a list of vectors of equal length:

Many operations work with matrices, like Dot, Transpose, and Det:

A rectangular array is represented by nested lists with consistent dimensions:

Many operations work on arrays of any depth, like Dot and Fourier:

The three-dimensional discrete Fourier transform:

Ragged arrays that are not rectangular can also be used:

Many structural functions will work with ragged arrays:

If the elements are at the same depth, you can use PadRight to make a rectangular array:

### Constructing Lists (5)

Range constructs a list consisting of a range of values:

Array constructs lists using a function:

When given multiple dimensions, matrices or deeper arrays are constructed:

Table constructs lists using an expression and an iterator:

When given multiple iterators, matrices and arrays can be constructed:

Functional commands like NestList create lists of the results:

To construct a list when the length is not known ahead of time, Sow and Reap are efficient:

Some trials of rolling a die until the same number comes up twice in a row:

### Listable Functions (3)

Matrix plus a vector adds the component of the vector to the rows of the matrix:

Function applied element-wise to a matrix:

Any function that has the Listable attribute will thread over lists element-wise:

### Operations on List Elements (5)

Apply makes the elements of a list the arguments of a function:

If you have a nested list, applying at level 1 gives a list f applied to the sublists:

Map applies a function to the elements of a list:

For a nested list, Map can apply f at any level or multiple levels:

Do, Product, Sum, and Table can iterate over a list:

Part can be used to get elements of lists:

You can get multiple parts by specifying a list of parts:

Or by using Span:

Use Outer to apply a function to elements of multiple lists:

### Lists as Finite Sets (2)

Complement, Union, and Intersection treat List as a set:

Construct various combinatorial structures using Subsets, Tuples, and IntegerPartitions:

### Lists as Control Structures (2)

### Lists of Rules (2)

A list of rules is returned as a solution by many solving commands:

You can use the values of the results with ReplaceAll:

When multiple solutions are possible, the result is a list of rule lists:

When a list of rule lists is used in ReplaceAll, you get a list of results:

Even if there is only one solution, the extra List is used for consistent structure:

### Lists of Data (3)

Lists are very good for holding data since the elements can be anything:

Data from a function sampled at points in two dimensions:

A piecewise polynomial that interpolates the data:

Plot the InterpolatingFunction:

## Generalizations & Extensions (2)

## Properties & Relations (1)

A SparseArray represents a list:

They are Equal:

They can be equivalently used in many commands:

They are not identical because the representation is different: