from @alanedelman:

We really should think carefully about how the transpose of a vector should dispatch the various A_*op*_B* methods. It must be possible to avoid new types and ugly mathematics. For example, vector'vector yielding a vector (#2472, #2936), vector' yielding a matrix, and vector'' yielding a matrix (#2686) are all bad mathematics.

What works for me mathematically (which avoids introducing a new type) is that for a 1-dimensional Vector v:

• v' is a no-op (i.e. just returns v),
• v'v or v'*v is a scalar,
• v*v' is a matrix, and
• v'A or v'*A (where A is an AbstractMatrix) is a vector

A general N-dimensional transpose reverses the order of indices. A vector, having one index, should be invariant under transposition.

In practice v' is rarely used in isolation, and is usually encountered in matrix-vector products and matrix-matrix products. A common example would be to construct bilinear forms v'A*w and quadratic forms v'A*v which are used in conjugate gradients, Rayleigh quotients, etc.

The only reason to introduce a new Transpose{Vector} type would be to represent the difference between contravariant and covariant vectors, and I don't find this compelling enough.