Linear form

In mathematics, a linear form is a special rule 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.

Imagine you have arrows pointing in different directions and lengths — these are vectors. A linear form is like a machine that looks at an arrow and tells you how much it "points" in a certain direction, giving you a single number.

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. When we fix a way to describe our vectors, these machines can be written down as rows of numbers. This helps mathematicians solve many problems.

Examples

The zero function 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.
• Finding the average: The average of a list of numbers can be found by adding all numbers and dividing 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.

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.

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.

(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 follows the rules of linearity.

Visualization

A linear functional can be imagined using its level sets. These are groups of vectors that give the same result. In three dimensions, these level sets look like flat, parallel planes. When we think about more dimensions, they become parallel hyperplanes. This way of visualizing linear functionals is sometimes shown in books about general relativity.

Applications

Linear forms have many useful applications in mathematics and physics. In numerical quadrature, linear forms help us estimate 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 shown using special spaces called Hilbert spaces, which are closely related to their dual spaces. For more details, see bra–ket notation.

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

Dual vectors and bilinear forms

A bilinear form is a way to combine two vectors to get a number. When this form is non-degenerate, it links the vector space to its dual space. This link helps 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 means changing the type of numbers used with a vector space.

If we begin with a vector space using complex numbers (written C), we can also think of it as a vector space using only real numbers (written R). This is called the "realification" of the original space.

How linear functions act can differ based on whether we use complex or real numbers. A complex linear function works with complex numbers, while a real linear function works only with real numbers. These differences matter in advanced parts of mathematics.

In infinite dimensions

See also: Continuous linear operator

When we work with very large spaces, called infinite-dimensional spaces, the idea of a linear form becomes more complex but also very useful. In these spaces, we often talk about continuous linear functionals. This means the way the functional changes is smooth and predictable.

One important idea is that in infinite dimensions, not every linear functional behaves the same way. Some of them are "continuous," meaning they don’t change suddenly. These continuous linear functionals have special properties that make them important in advanced mathematics, especially in areas called analysis and functional analysis. They can help us understand certain parts of our space.