How do you define a linear map?
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping. between two vector spaces that preserves the operations of vector addition and scalar multiplication.
What is a k linear map?
A K-linear map (also K-linear function, K-linear operator, or K-linear transformation) is a morphism in K-Vect (or K-Mod), that is a homomorphism of vector spaces (or modules). (It is an easy exercise that this one identity is enough to ensure that T preserves all linear combinations.)
What is map in cryptography?
From Wikipedia, the free encyclopedia. A cryptographic -multilinear map is a kind of multilinear map, that is, a function such that for any integers and elements , , and which in addition is efficiently computable and satisfy some security properties.
Is a tensor a multilinear map?
A tensor of type (r,s) on V is a multilinear map T:Vr×(V∗)s→R.
How can you tell if a map is linear?
It is simple enough to identify whether or not a given function f(x) is a linear transformation. Just look at each term of each component of f(x). If each of these terms is a number times one of the components of x, then f is a linear transformation.
How is linear transformation defined?
A linear transformation is a function from one vector space to another that respects the underlying (linear) structure of each vector space. A linear transformation is also known as a linear operator or map.
What is the meaning of multilinear?
adjective. having or involving several lines.
What is Hom VW?
linear-algebra vector-spaces linear-transformations. WTS: Hom(V,W) which is the set of all linear maps is a vector space.
What is chaotic map?
In mathematics, a chaotic map is a map (= evolution function) that exhibits some sort of chaotic behavior. Maps may be parameterized by a discrete-time or a continuous-time parameter. Chaotic maps often occur in the study of dynamical systems. Chaotic maps often generate fractals.
Is the determinant multilinear?
The determinant is multilinear in the rows. This means that if we fix all but one column of an n × n matrix, the determinant function is linear in the remaining column.
Are linear maps smooth?
Section 1, #5 Show that every k-dimensional vector subspace V of RN is a manifold diffeomorphic to Rk, and that all linear maps on V are smooth. Thus φ is a diffeomorphism. The fact that all linear maps on V are smooth also follows from the next exercise.
What kind of map is a multilinear form?
If the codomain of a multilinear map is the field of scalars, it is called a multilinear form. Multilinear maps and multilinear forms are fundamental objects of study in multilinear algebra. If all variables belong to the same space, one can consider symmetric, antisymmetric and alternating k-linear maps.
What do you call a multilinear map of k variables?
More generally, a multilinear map of k variables is called a k-linear map. If the codomain of a multilinear map is the field of scalars, it is called a multilinear form.
Which is an example of a multilinear function?
For example, any inner product on a vector space is a multilinear map, as is the cross product of vectors in . The determinant of a matrix is an alternating multilinear function of the columns (or rows) of a square matrix. If is a Ck function, then the th derivative of at each point in its domain can be viewed as a symmetric -linear function .
Is the determinant of a matrix a multilinear function?
The determinant of a matrix is an alternating multilinear function of the columns (or rows) of a square matrix. If is a Ck function, then the th derivative of at each point in its domain can be viewed as a symmetric -linear function . The tensor-to-vector projection in multilinear subspace learning is…