Linear subspace

The kernel of 5) is a linear subspace of B,), which we use to define an equivalence relationship on... 

Ellingsrud and Strpmme have constructed the cell decomposition of P using the following results of [Bialynicki-Birula (1),(2)]. Let X be a smooth projective variety over k with an action of the multiplicative group Gm. We will denote this action by Let x X be a fixed point of this action. Let T x C Tx,x be the linear subspace on which all the weights of the induced action of Gm are positive. 

In section 3.2 we consider the varieties of higher order data D X). Their definition is a generalisation of that of D X). We show that only the varieties of third order data of curves and hypersurfaces are well-behaved, i.e. they are locally trivial bundles over the corresponding varieties of second order data with fibre a projective space. In particular D X) is a natural desingularisation of. Then we compute the Chow ring of these varieties. As an enumerative application of the results of chapter 3 we determine formulas for the numbers of second and third order contacts of a smooth projective variety X C Pn with linear subspaces of P. ... 

If N = (n 1), then the number of n — l)th order contacts with 2-codimensional linear subspaces m Pai is... 

E has a one-dimensional intersection with a fixed 2-codimensional linear subspace... 

So theorem 4.4.6 describes the Chow ring of Cop t(P ) in terms of classes describing the position of subschemes relative to linear subspaces of Pd. 

The semidefinite program we are considering is formulated in terms of the symmetric matrix H and the linear subspace of symmetric matrices S. For the... 

Exercise 2.20 (Used in Section 3.4) Show that if T is a linear transformation with domain V, and W is any linear subspace ofV. then the restriction T -w of T to W is a linear transformation. 

However, the notion of a linear subspace descends to projective space. Definition 10.2 IfW is a linear subspace of V, we define... 

Such a subset ofPfV) is called a linear subspace of P(V ). 

Not every linear-subspace-preserving function on projective space descends from a complex linear operator. However, when we consider the unitary structure in Section 10.3 we find an imperfect but still useful converse — see Proposition 10.9. 

It is useful to know that these spin representations have no multiplicities. Recall from Proposition 10.1 that multiplicities of eigenvalues on linear spaces correspond to dimensions of linear subspaces of fixed points of the projectivization. For example, the points [1 0], [0 1] are the only fixed points of pi (Xe) (for d ttZ). They form a basis of (C ). Similarly, in any spin representation the eigenstates of rotations around any one axis have isolated fixed points that form a basis for the state space. 

To extend this result to projective space of arbitrary finite dimension we will need the technical proposition below. Since addition does not descend to projective space, it makes no sense to talk of linear maps from one projective space to another. Yet something of linearity survives in projective space subspaces, as we saw in Proposition 10.1. The next step toward our classification is to show that physical symmetries preserve finite-dimensional linear subspaces and their dimensions. 

Proposition 10.9 Suppose U and V are complex scalar product spaces and S P((/) P(y) preserves the absolute value of the bracket. Suppose U is finite-dimensional. There is a linear subspace V5 of V such that [V5] is the image of[U under S. Furthermore, dim V5 = dim U. 

By definition, this Vs is a linear subspace of V. Since 5 preserves brackets, V is orthogonal to Vo. Hence... 

Because S preserves linear subspaces and dimensions (by Proposition 10.9), the image X of W under 5 is two-dimensional and is spanned by the lines 5([u -H ac ]) and 5([u -F 5e ]). Hence, the line N([wa ]) = [w al, which lies in must also lie in the subspace X, Since a 0, the subspace X does not lie entirely in En hence the two-dimensional subspace X must intersect the n-dimensional subspace in a one-dimensional subspace. The intersection must be the subspace [wa,a] on the other hand, we can calculate the intersection explicitly from the basis of X. Since v and v are linearly independent, we can construct a nonzero element of n X ... 

Recall from Proposition 3.5 that given any finite-dimensional linear subspace W of a scalar product space V, there is an orthogonal projection fliv on V whose kernel is VP- -. Note that the expression giving the probability does not depend on the choice of vector in the equivalence class [v]. 

All conservation laws together define a linear subspace of lattice points in the total state space. The accessible subspace lies in this subspace and usually is identical with it, but not necessarily so. A counterexample would be... 

When there are more macroscopic observables B,C,... the process can be continued. The end result is a collection of coarse-grained observables //, A, 5, C, ..., which all commute with one another. The Hilbert space H is decomposed in linear subspaces that are common eigenspaces of these observables. We shall call these subspaces phase cells and indicate them with a single label J. They correspond to definite values of the coarse-grained variables, which we shall now denote by Ej, AJy BJy Cj,. These phase cells are the macrostates. 

---