In order to prove the following Theorem 3.2, which characterizes the form of the linear map
preserving the left spectrum of 2 x 2 quaternion matrices, we need the following lemmas.
From  (Example 2.13) we know that if one considers the linear map
[alpha]: A [right arrow] A given by
* For an arbitrary vector space W, we denote by Hom(W, C) the set of all linear maps
from W to C.
A bilinear map is a function [phi] : U [right arrow] D such that for any a [member of] A the map b [??] [phi](a, b) is linear map
from B to D, and for any b [member of] B the map a [??] [phi](a, b) is linear map
from A to D.
This minimization problem is always uniquely solvable, and as the optimal linear map
minimizing (3.2) we obtain
We shall now show that each [PHI] [member of] M defines a continuous linear map
from B into B.
Understanding the correspondence between matrices and affine transformations can help to clarify the meaning of such key notions from Linear algebra and Geometry such as the linear map
, affine map, image and kernel of an affine map, composition of affine maps and transformation of the coordinate system.
A linear map
T: C(X) [right arrow] C(Y) is called Separating (, p.2) if fg [equivalent to] 0 implies TfTg [equivalent to] 0.
The output graph is a piecewise linear map
, which is possibly make as a smooth curve by standard methods or by new one.
If you are someone who prefers a maze instead of a linear map
, this will be your thing.
Let S be a subset of a vector space over [Z.sub.2] and choose a random linear map
to a smaller vector space R.
Recall from Part I that we defined for any abelian F-variety A a number [rank.sub.[Delta]](A) called its [Delta]-rank; it is simply the rank of the linear map