Tensors

Tensors in linear algebra are mathematical objects that generalize scalars, vectors, and matrices to higher dimensions.

At their core, tensors are multilinear maps or multilinear functions. They can be understood as objects that remain invariant under coordinate transformations while representing physical or mathematical quantities. This invariance property makes them particularly powerful for describing physical laws and mathematical relationships that should remain consistent regardless of the chosen coordinate system.

The concept of tensors emerged in the 19th century through the work of mathematicians studying geometry and physics. Notably, Ricci-Curbastro and his student Tullio Levi-Civita developed tensor calculus, which later became instrumental in Einstein's formulation of general relativity. Today, tensors have transcended their geometric origins to become essential tools in fields ranging from continuum mechanics to machine learning.

Consider why we need tensors by examining the progression of mathematical structures:

• Scalars (0th-order tensors) represent magnitude alone

• Vectors (1st-order tensors) add direction to magnitude

• Matrices (2nd-order tensors) describe linear transformations and relationships between vectors

• Higher-order tensors capture more complex multilinear relationships

This hierarchy reveals tensors as a natural extension of familiar mathematical objects, each level adding new capabilities for representing increasingly complex relationships.