Linear form

In mathematics, a linear form is a special kind of rule or function that takes a vector and gives back a number. It is also called a linear functional, a one-form, or a covector. This idea is important in many areas of math because it helps us understand how vectors relate to numbers in a straight-line way.

Imagine you have a set of arrows pointing in different directions and lengths — these are your vectors. A linear form is like a machine that looks at one of these arrows and tells you how much it "points" in a certain direction, giving you a single number as a result. This machine follows special rules: if you add two arrows together first and then measure them, it’s the same as measuring each arrow separately and adding those numbers together. The same goes for stretching or shrinking an arrow before measuring it.

The collection of all these measuring machines for a given set of vectors forms its own vector space, called the dual space. This dual space has its own rules for adding and scaling these machines, just like the original vectors do. When we fix a way to describe our vectors (called a basis), these machines can be written down as flat rows of numbers, and their results are found by multiplying these rows by the column of numbers that describe a vector. This connection to matrices helps mathematicians solve many problems in geometry, physics, and engineering.

Examples

The zero function, which sends every vector to zero, is a simple example of a linear functional. Other linear functionals include:

Selecting an element from a list: For a list of three numbers, we can pick just the second number using a special tool made for this purpose.
Finding the average: The average of a list of numbers can be found using another tool that adds all numbers and divides by how many there are.
Calculating present value: In money matters, we can find the current worth of future cash flows using a specific method that considers the time value of money.

Linear functionals in Rⁿ

In basic math, we can think of vectors as lists of numbers. For each list of numbers, there is a way to create a linear functional by using a rule that combines these numbers in a straight-line fashion.

Trace of a square matrix

The trace of a square matrix is the sum of the numbers along the main diagonal. This trace acts like a linear functional because it follows specific rules when we add matrices or multiply them by numbers.

(Definite) Integration

In more advanced math, linear functionals appear when we study spaces of functions. One common example is integration, where we add up values of a function over an interval. This process is linear because it behaves nicely when we add functions or scale them by numbers.

Evaluation

For polynomials — functions made from adding and multiplying variables and numbers — we can evaluate them at specific points. This evaluation is a linear functional because it also follows the rules of linearity when we add polynomials or scale them.

Visualization

In simple terms, a linear functional can be pictured using its level sets, which are the groups of vectors that give the same result. In three dimensions, these level sets look like a set of flat, parallel planes. When we think about more dimensions, these level sets become parallel hyperplanes. This way of visualizing linear functionals is sometimes shown in books about general relativity, like Gravitation by Misner, Thorne & Wheeler (1973).

Applications

Linear forms have many useful applications in mathematics and physics. In numerical quadrature, linear forms help us approximate the area under a curve by using specific points and combining their values.

In quantum mechanics, linear forms are important because they help describe the states of quantum systems. These systems are often represented using special spaces called Hilbert spaces, which have a close relationship with their dual spaces. For more details, see bra–ket notation.

In the study of generalized functions, certain types called distributions can also be understood as linear forms acting on test functions.

Dual vectors and bilinear forms

A bilinear form is a special way to combine two vectors to get a number. When this form is non-degenerate, it creates a connection between the vector space and its dual space. This connection lets us find a special vector in the dual space for each vector in the original space, called the dual vector.

In spaces with infinite dimensions, like Hilbert spaces, similar ideas work thanks to the Riesz representation theorem. This theorem helps us understand how vectors and their duals relate in these more complex spaces.

Relationship to bases

When we look at a special kind of math space called a vector space, we can think of it like a grid with points that we can move along in straight lines. A linear form is a rule that takes any point in this space and gives us a number, following certain straight-line rules.

If the space has a certain number of basic points (like the corners of a shape), we can match each basic point with a special rule. These matching rules form what’s called the dual basis. They help us understand how the space behaves and can be used to find out important details about any point in the space by using simple multiplication and addition.

Main article: Schauder basis

Change of field

The change of field in linear algebra refers to adjusting the type of numbers (or "scalars") used with a vector space. When we start with a vector space over the complex numbers (denoted C), we can also view it as a vector space over the real numbers (denoted R) by only allowing real number scalars. This creates what is called the "realification" of the original space.

Linear functionals behave differently depending on whether we use complex or real scalars. A complex linear functional takes complex inputs and gives complex outputs, while a real linear functional works only with real inputs and outputs. These differences are important in advanced mathematics, especially in areas like functional analysis and Hilbert spaces.