Multilinear algebra facts for kids
Multilinear algebra is a part of mathematics. It builds on something called linear algebra. Linear algebra deals with simple things like vectors. Multilinear algebra takes this a step further. It looks at more complex ideas like p-vectors and multivectors. It uses a system called Grassmann algebra to do this.
How it Started
In a vector space, we usually focus on single vectors. But a mathematician named Hermann Grassmann thought differently. He believed it was important to look at pairs, triplets, and even more complex groups of vectors. These are called multi-vectors. Thinking this way makes the math much bigger and more detailed. For example, the way we figure out the determinant of a matrix comes from multilinear algebra. It also helps scientists understand how materials react to stress.
The word "tensor" came about to describe these special elements in multilinear algebra. This new way of thinking made multilinear algebra very important in advanced math. Grassmann first wrote about these ideas in 1844. But it took a long time for people to accept it. Regular linear algebra was already quite challenging for many.
Later, ideas from multilinear algebra were used in multivariate calculus. This is a type of calculus that deals with many variables. It also helps with understanding manifolds, which are like curved spaces. For example, small changes in single-variable calculus become differential forms in multivariate calculus. Exterior algebra is used to work with these forms.
After Grassmann, other mathematicians like Victor Schlegel and Elwin Bruno Christoffel added to the field. But big steps forward came from Gregorio Ricci-Curbastro and Tullio Levi-Civita. They developed something called "absolute differential calculus." This was a key part of multilinear algebra.
In 1915, Albert Einstein used these ideas in his theory of general relativity. He used multilinear algebra and tensors to explain how planets move. This made multilinear algebra a very important tool in physics.
In 1958, a group of mathematicians called Nicolas Bourbaki wrote a book about multilinear algebra. It explained important topics like bilinear functions and tensor products.
Applications
Multilinear algebra is used in many different areas. Some of these include:
- Understanding how tensors are used in physics.
- Working with geometric algebra, which combines geometry and algebra.
- Studying Clifford algebra, another advanced math topic.
- Learning about spinors, which are used in quantum physics.
- Dealing with hypercomplex numbers, which are like expanded versions of regular numbers.
See also
In Spanish: Álgebra multilineal para niños
