# Tensor vector dot product vs matrix vector dot product

If $$A$$ is a matrix and $$B$$ is a tensor (for example 3 by 3/rank 2, and with the same components) and $$v$$ is a 3 by 1 vector,

• Is there any difference between $$A.v$$ and $$B.v$$(in terms of the formula to compute them), where $$.$$ is the dot product. Are they both a normal "matrix multiplication" which result a 3 by 1 vector?

• Do you understand the difference between a "matrix" and a "tensor"? It is the same as the difference between a "triple of numbers" and a vector. A tensor can be represented by a matrix in a given coordinate system. If we change the coordinate system the matrix may change but the tensor is the same, just represented by a different matrix. Jun 25, 2021 at 19:26
• any of your matrices are used as $Av$ to give you a linear transformation but also $v^{\top}Aw$ to have a bilinear map into the scalar field Jun 25, 2021 at 20:41

If $$A$$ is used to represent a linear transformation then ones could use $$w^k=A_s{}^kv^s,$$ to get $$n$$ quantities (since $$1\le k\le n$$) for the components of $$w$$, from those of $$v$$.
Or for a bilinear map $$B$$, where two vectors $$u,v$$ are paired via $$B_{st}u^sv^t,$$ to assign a number.