[Richard777 (OP)]

The product of a tensor and a vector is a different vector.
This suggests that a tensor may be represented as a ratio of two vectors.
Is this true?

[PmbPhy]

The product of a tensor and a vector is a different vector.
This suggests that a tensor may be represented as a ratio of two vectors.
Is this true?

No. A tensor changes the components of a vector in one frame to that of another frame i.e. V' = TV where V is a matrix, not a of two vectors..