Transformations and tensors

Let us start from the m-dimensional space f i and consider the basis change (2.2.16), namely 

The components c of any given vector (i.e. a general element of the vector space) must then be changed according to (2.2.18), becoming 

In the systematic study of such transformations it is usual to adopt the language of tensor algebra, expressing both transformation rules in a common form for ease of comparison. If we put R = then we may write 

The two transformations in (10.2.2) are said to be contragredient, the first (typical of basis vectors) being covariant and the second (typical of vector components) being contravariant. The relationship is clearly reflexive in the sense that if we put R = U then R = tl and we can just as well write (10.2.2) as 

When U in (10.2.4) is a general non-singular mxm matrix the infinite set U forms a matrix group, the full linear group in m dimensions, denoted by GL(m). If the matrices are chosen to be unitary (thus leaving invariant any Hermitian scalar product, as we know from Section 2.2) then we obtain the unitary group U(m) and in this case the matrices of the covariant transformation in (10.2.4b) are 

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