A linear mapping from a vector space to a field of scalars. In other words, a linear function which acts upon a vector resulting in a real number (scalar) α:V⟶R
Tensors Tensor Product
A mapping from V→W that preserves the operations of addition and scalar multiplication. Also known as # Linear Map Linear Transformation Linear Function
A function of several variables that is linear, separately, in each variable. A multilinear map of one variable is a standard linear mapping.
The space of all linear functionals f:V→R, noted as V∗ The dual space has the same dimension as the corresponding vector space or, given a space V, with bases (v1,…,vn), there exists a dual space V∗ with a dual basis (v∗1,…,v∗n).
The space of all linear functionals f:V→R, noted as V∗ The dual space has the same dimension as the corresponding vector space or, given a space V, with bases (v1,…,vn), there exists a dual space V∗ with a dual basis (v∗1,…,v∗n).
As a linear representation # A tensor can be represented as a vector of x-number of dimensions. Basically, a generalization on top of scalars, vectors, and matrices. The specific “flavor” of the tensor (i.e. is it a scalar, vector, or matrix) is clarified by referring to the tensor’s “rank”. For instance; a rank 0 tensor is a scalr, rank 1 tensor is a one-dimensional vector, a rank 2 tensor is a two-dimensional vector (2x2 matrix), etc. ...
a basis for an n-dimensional vector space V is any ordered set of linearly independent vectors (e1,e2,…,en) An arbitrary vector x in V can be expressed as a linear combination of the basis vectors: x=n∑i=1eixi
An orthonormal basis is a basis where all the vectors are one unit long and all perpendicular to each other (e.g. the Cartesian plane)