TITLE: How to visualize a second-order tensor in a rectangular coordinate system?
QUESTION [5 upvotes]: I can visualize a first-order tensor as a segment (a vector), but I'm not sure how to visualize a second-order tensor. The book that I'm trying to study is "Vector and tensor analysis with applications" (A. Borisenko and I. Tarapov)

REPLY [2 votes]: For a vector space $V$ over a scalar field ${\Bbb F}$, you can see it as a bilinear transformation
$$B:V\times V\longrightarrow{\Bbb F}$$
which allow you to impose a metric in $V$. 
If we consider that $V$ is spawned by a basis $\{b_i\}$,
then it is necessary that the matrix 
$$[B]=[B(b_i,b_j)]$$
complies certain conditions.