By v. Jowett
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Extra info for Aristoteles, Politica BB
This projection produces an image of the axis at a point in the dual space. The concept can be extended to the product of a matrix with another matrix. In this case we can conceive of the latter as a set of axes, each of which produces image points in the dual space. In the special case when this matrix has only two columns, the product can be regarded as a projection of a pattern of points upon the plane formed by the two axes. As a result one obtains two image points (one for each axis that defines the plane of projection) in the dual space.
The word projection is reserved for operations which map points or axes in the same space . The imaging of v in S^ into s in S"" is represented geometrically in Fig. 9a. Note that the patterns of points P"^ and P^ are represented schematically by elliptic envelopes. g. Xy, in 5" which can be projected upon an axis u and which results into a point I (not to be confounded with the sum vector 1) in the complementary (or dual) space 5^ (Fig. ,/7 /. =xTu Using the same argument as above, we can see that the product is of dimension p, since the matrix X^ has dimensions pxn and the vector u possesses dimension n.
Note that the patterns of points P"^ and P^ are represented schematically by elliptic envelopes. g. Xy, in 5" which can be projected upon an axis u and which results into a point I (not to be confounded with the sum vector 1) in the complementary (or dual) space 5^ (Fig. ,/7 /. =xTu Using the same argument as above, we can see that the product is of dimension p, since the matrix X^ has dimensions pxn and the vector u possesses dimension n. Hence, the vector can be imaged as a single point in the dual space S^.
Aristoteles, Politica BB by v. Jowett