Abstract
Let V be a vector space over a field \(\mathbb{k}\). A linear map \(\xi: V \rightarrow \mathbb{k}\) is called a covector or linear form on V. The covectors on V form a vector space, denoted by and called the dual space to V. We have seen in Sect. 6.3.2 on p. 136 that every linear map is uniquely determined by its values on an arbitrarily chosen basis. In particular, every covector ξ ∈ V ∗ is uniquely determined by numbers \(\xi (e) \in \mathbb{k}\) as e runs trough some basis of V. The next lemma is a particular case of Proposition 6.2 on p. 137. However, we rewrite it here once more in notation that does not assume the finite-dimensionality of V.
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Notes
- 1.
Also linear functional.
- 2.
Not necessarily finite.
- 3.
See Example 6.8 on p. 129.
- 4.
See formula (6.15) on p. 129.
- 5.
Recall that all but a finite number of the λ e equal zero.
- 6.
Meaning that it does not depend on any extra data such as the choice of basis.
- 7.
For infinite-dimensional spaces, this is not true, as we saw in Example 7.3.
- 8.
See the comments before formula (6.11) on p. 126.
- 9.
See Example 6.4 on p. 125.
- 10.
That is, between k- dimensional and (dim V − k)- dimensional subspaces for each k = 0, 1, … , dim V.
- 11.
Possibly infinite-dimensional.
- 12.
This theorem is known as the Rouché–Fontené theorem in France, the Rouché–Capelli theorem in Italy, the Kronecker–Capelli theorem in Russia, and the Rouché–Frobenius theorem in Spain and Latin America.
- 13.
It is called the ring of linear differential operators of finite order with constant coefficients.
- 14.
Here \(\mathbb{Q}[D]/\left (D^{n+1}\right )\) means the quotient of the polynomial ring \(\mathbb{Q}[D]\) by the principal ideal spanned by D n+1. The vector space \(\mathbb{Q}[x]_{\leqslant n}\) can be thought of as the space of solutions of the linear differential equation D n+1 y = 0 in the unknown function y. Both \(\mathbb{Q}[D]/\left (D^{n+1}\right )\) and \(\mathbb{Q}[x]_{\leqslant n}\) are considered just vector spaces over \(\mathbb{Q}\).
- 15.
Recall that \(\Phi f\stackrel{\text{def}}{=}a_{0}f + a_{1}Df + a_{2}D^{2}f +\, \cdots \, + a_{n}D^{n}f\) for \(\Phi = a_{0} + a_{1}D + \cdots + a_{n}D^{n}\) (compare with Sect. 4.4 on p. 88).
- 16.
Whose eigenvalue may depend on the vector.
- 17.
A matrix A = (a ij ) is diagonal if a ij = 0 for all i ≠ j.
- 18.
Considered as a vector space over \(\mathbb{k}\).
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Gorodentsev, A.L. (2016). Duality. In: Algebra I. Springer, Cham. https://doi.org/10.1007/978-3-319-45285-2_7
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