§ 1. Classical Differential Calculus
135
i.e. a linear function of h i when the other variables are held fixed, (ii) it is
alternating, i.e. changes sign when two of the variables h i and h j are permuted. Choosing a basis (a i ) for E and assuming D(a 1 , . . . , a n ) = 1, which
determines D, the number D(h 1 , . . . , h n ) is the determinant of the h i with
respect to the given basis. It is non-zero if and only if the h i are linearly independent, i.e form a basis of E. The formula for calculating D(h 1 , . . . , h n )
explicitly from the coordinates of the h i can be found everywhere.
Given a linear map u : E −→ E, the determinant of u is the determinant
of the vectors u(a i ), where (a i ) is an arbitrary basis for E; it does not depend
on it. Its basic properties are (i)
D [u (h 1 ) , . . . , u (h n )] = det(u)D (h 1 , . . . , h n )
(1.3’)
for all h i ∈ E, and so
det(u ◦ v) = det(u) det(v)
(1.3”)
for all u, v : E −→ E, (ii) u is injective (or what amounts to the same in
finite dimension, surjective or bijective) if and only if det(u) = 0.
More generally, consider a linear map u : E −→ F , where E and F may
have different dimensions, and let r be its rank, i.e. the dimension of the
subspace u(E) of F . Let A = (u
j
i ) be the matrix of u with respect to two
arbitrary bases of E and F . Square matrices of order ≤ min[dim(E), dim(F )]
can be extracted from it by arbitrarily choosing the same number of rows and
columns of A. Having said that, r is the largest integer for which a square
matrix of order r and non-zero determinant can be extracted from A.
Finally, note that instead of “ finite-dimensional vector space ”, we will
mostly use the expression real or complex Cartesian space when K = R or
C. These are the only cases occurring in classical analysis.
(ii) Tensor notation. When Albert Einstein began his work in general
relativity, he learnt the hard way, with the help of his mathematician friends
who had read the Italian literature on differential geometry, not to mix up
vectors and linear functionals (and, more generally, to distinguish what are
called covariant tensors from contravariant ones – see below), despite the fact
that a vector has as many components as a linear functional has coefficients.
Indeed, if there is basis change, by (1), the coefficients of a linear functional
undergo the same linear transformation as the basis vectors, whereas the coordinates of a vector undergo a “ contragredient ” one. This term is here used
in in sense in which it is in the context of square matrices: the inverse of the
transpose. For example in the simplest case, where the basis (a 1 , . . . , a n ) is replaced by the basis (t 1 a 1 , . . . , t n a n ), with scalars t i = 0, the u i are multiplied
and the h
i divided by the t i . Hence, if, with respect to a particular basis, a
vector and a linear functional appear to coincide because they have the same
coordinates, this is not the case with respect to others; equating them has
no physical and mathematical meaning. Anyhow, they are not objects of the
same nature: a function defined on a set is not an element of this set.
135
i.e. a linear function of h i when the other variables are held fixed, (ii) it is
alternating, i.e. changes sign when two of the variables h i and h j are permuted. Choosing a basis (a i ) for E and assuming D(a 1 , . . . , a n ) = 1, which
determines D, the number D(h 1 , . . . , h n ) is the determinant of the h i with
respect to the given basis. It is non-zero if and only if the h i are linearly independent, i.e form a basis of E. The formula for calculating D(h 1 , . . . , h n )
explicitly from the coordinates of the h i can be found everywhere.
Given a linear map u : E −→ E, the determinant of u is the determinant
of the vectors u(a i ), where (a i ) is an arbitrary basis for E; it does not depend
on it. Its basic properties are (i)
D [u (h 1 ) , . . . , u (h n )] = det(u)D (h 1 , . . . , h n )
(1.3’)
for all h i ∈ E, and so
det(u ◦ v) = det(u) det(v)
(1.3”)
for all u, v : E −→ E, (ii) u is injective (or what amounts to the same in
finite dimension, surjective or bijective) if and only if det(u) = 0.
More generally, consider a linear map u : E −→ F , where E and F may
have different dimensions, and let r be its rank, i.e. the dimension of the
subspace u(E) of F . Let A = (u
j
i ) be the matrix of u with respect to two
arbitrary bases of E and F . Square matrices of order ≤ min[dim(E), dim(F )]
can be extracted from it by arbitrarily choosing the same number of rows and
columns of A. Having said that, r is the largest integer for which a square
matrix of order r and non-zero determinant can be extracted from A.
Finally, note that instead of “ finite-dimensional vector space ”, we will
mostly use the expression real or complex Cartesian space when K = R or
C. These are the only cases occurring in classical analysis.
(ii) Tensor notation. When Albert Einstein began his work in general
relativity, he learnt the hard way, with the help of his mathematician friends
who had read the Italian literature on differential geometry, not to mix up
vectors and linear functionals (and, more generally, to distinguish what are
called covariant tensors from contravariant ones – see below), despite the fact
that a vector has as many components as a linear functional has coefficients.
Indeed, if there is basis change, by (1), the coefficients of a linear functional
undergo the same linear transformation as the basis vectors, whereas the coordinates of a vector undergo a “ contragredient ” one. This term is here used
in in sense in which it is in the context of square matrices: the inverse of the
transpose. For example in the simplest case, where the basis (a 1 , . . . , a n ) is replaced by the basis (t 1 a 1 , . . . , t n a n ), with scalars t i = 0, the u i are multiplied
and the h
i divided by the t i . Hence, if, with respect to a particular basis, a
vector and a linear functional appear to coincide because they have the same
coordinates, this is not the case with respect to others; equating them has
no physical and mathematical meaning. Anyhow, they are not objects of the
same nature: a function defined on a set is not an element of this set.