C*-algebras and their automorphism groups by Gert Kjaergard Pedersen
By Gert Kjaergard Pedersen
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Ai are all different, . is in L(, n). (Hint: The determinant of this matrix is a so-called Vandermonde determinant. ) i>i 2. Prove that if Ll. = is a field and (a) e L(, n), then the transposed matrix (a)' e L(, n). 3. Prove the following converse of Ex. , 2), then Ll. is a field. 25 FINITE DIMENSIONAL VECTOR SPACES 4. ] -: -5 1 7 9. Factor spaces. Any subspace to of ~ is, of course, a subgroup of the additive group~. Since ~ is commutative, we can define the factor group lR = ~/t0.
J. Hence dim [XI, X2, ... , xrl = dim [YI, Y2, .. ·,Yrl· In a similar fashion we define the column rank of (a). Here we introduce a right vector space m' of r dimensions with basis (el', e2', "', e/). Then we define the column rank of (a) to be the rank of the set (Xl" X2', ... , x n ') where x/ = "1;e/ aji. The x/ are called column vectors of (a). We shall prove in the next chapter that the two ranks of a matrix are always equal. In the special case Ll. = cf> a field (commutative) this equality can be established by showing that these ranks coincide with still another rank which can be defined in terms of determinants.
Zi = L Zj e @5i j r'i + + ... + + ... + n (e;l + ... + e;i-l + e;i+l + ... + e;r). Hence Zi = 0 and Yi = Y/. The converse of this result holds also; for if (9) fails for some i, then there is a vector Zi ~ 0 in this intersection. Thus Zi = L Zh and we have two distinct j r"i representations of this element as a sum of elements out of the spaces elk. We have therefore proved Theorem 10. A necessary and sufficient condition that the spaces e;l, e;2, ... , e;r be independent is that every vector in @5 = e;l e;2 e;r have a unique representation in the form ~Yi, Yi in @5i.