Can represent a symmetric bilinear form as shown at matrix representation of a symmetric bilinear form, or a quadratic form.
The definition implies that this is also a symmetric matrix.
The dot product is a positive definite matrix, and so we see that those will have an important link to familiar geometry.
WTF is a skew? "Antisymmetric" is just such a better name! And it also appears in other definitions such as antisymmetric multilinear map.
Undecidable Diophantine equation example by 
Ciro Santilli 35 Updated 2025-02-26 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-02-26 +Created 1970-01-01mathoverflow.net/questions/11540/what-are-the-most-attractive-turing-undecidable-problems-in-mathematics/103415#103415 provides a specific single undecidable Diophantine equation.
where:
- : matrix in the old basis
- : matrix in the new basis
- : change of basis matrix
Change of basis between symmetric matrices by Ciro Santilli 35 Updated 2025-02-26 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-02-26 +Created 1970-01-01When we have a symmetric matrix, a change of basis keeps symmetry iff it is done by an orthogonal matrix, in which case:
Every vector space is defined over a field.
E.g. in , the underlying field is , the real numbers. And in the underlying field is , the complex numbers.
Any field can be used, including finite field. But the underlying thing has to be a field, because the definitions of a vector need all field properties to hold to make sense.
Elements of the underlying field of a vector space are known as scalar.
A member of the underlying field of a vector space. E.g. in , the underlying field is , and a scalar is a member of , i.e. a real number.
Because a tensor is a multilinear form, it can be fully specified by how it act on all combinations of basis sets, which can be done in terms of components. We refer to each component as:where we remember that the raised indices refer dual vector.
Explain it properly bibliography:
- www.reddit.com/r/Physics/comments/7lfleo/intuitive_understanding_of_tensors/
- www.reddit.com/r/askscience/comments/sis3j2/what_exactly_are_tensors/
- math.stackexchange.com/questions/10282/an-introduction-to-tensors?noredirect=1&lq=1
- math.stackexchange.com/questions/2398177/question-about-the-physical-intuition-behind-tensors
- math.stackexchange.com/questions/657494/what-exactly-is-a-tensor
- physics.stackexchange.com/questions/715634/what-is-a-tensor-intuitively
A linear map can be seen as a (1,1) tensor because:is a number, . is a dual vector, and W is a vector. Furthermoe, is linear in both and . All of this makes fullfill the definition of a (1,1) tensor.
Pinned article: ourbigbook/introduction-to-the-ourbigbook-project
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