row/column notation in dual space

Hi guys,

I have a problem with the row/column notation in dual spaces.
I will illustrate the problem with an example.

Lin. mapping F from V -> W. V is 2-dim and W is 3-dim. K is the Field.
For simplicity we assume that K is the Reals R.
The column (!) 1 /cr 2 is the coordinate vector in K^2 of an element v1
in V. (/cr means carriage return)
We assume F to be the 3x2 matrix 1 & 2 /cr 3 & 4 /cr 5 & 6
F working on the coord. vector in K^2 gives the 3-dim coord. vector 5
/cr 11 /cr 17 in K^3 of the vector w1.
Sofar no problems.

Let V* en W* be the dual spaces of V and W. Let f belong to W*
I assume the coordinate vector of f to be the row (!!) (1,2,3)
When f works on de coord. vector w1 we get the scalar 5+22+51=78 So far
so good.

The adjoint of the mapping F is F* and is the 2x3 matrix 1 & 3 & 5 /cr 2
& 4 & 6
Now comes my problem. How can I get F* to work on f because I can't let
a 2x3 matrix work on a 1x3 matrix (the row of f)



Thanks in advance,
Peter
.