Vector space subspace

Linear operators. Let X and Y be normed vector spaces and T be a subspace of the space X. If to each vector x V there corresponds by an... 

The set of unit vectors of dimension n defines an n-dimensional rectangular (or Cartesian) coordinate space 5 . Such a coordinate space S" can be thought of as being constructed from n base vectors of unit length which originate from a common point and which are mutually perpendicular. Hence, a coordinate space is a vector space which is used as a reference frame for representing other vector spaces. It is not uncommon that the dimension of a coordinate space (i.e. the number of mutually perpendicular base vectors of unit length) exceeds the dimension of the vector space that is embedded in it. In that case the latter is said to be a subspace of the former. For example, the basis of 5 is ... 

A vector space Lm is said to be a subspace of another vector space Ln if every vector of Lm is also contained in Ln. Lm is called a proper subspace of Ln if the vectors of Lm do not exhaust the space Ln. 

The vector space Ln, which is invariant under G, may contain a proper subspace which is also invariant under G. In such a case, Lm is an invariant subspace of Ln under G, and the space Ln is said to be reducible under G. 

Any 37/-dimensional Cartesian vector that is associated with a point on the constraint surface may be divided into a soft component, which is locally tangent to the constraint surface and a hard component, which is perpendicular to this surface. The soft subspace is the /-dimensional vector space that contains aU 3N dimensional Cartesian vectors that are locally tangent to the constraint surface. It is spanned by / covariant tangent basis vectors... 

If a subset W of a vector space V satisfies the definition of a vector space, with addition and scalar multiplication defined by the same operation as in V, then W is called a vector subspace or, more succinctly, a subspace of V. For example, the trivial subspace 0 is a subspace of any vector space. 

A more interesting example involves the vector space P3 of complex-coefficient polynomials in three variables. Let H denote the subset of P3 containing only harmonic polynomials, i.e., only polynomials p in three variables satisfying = 0. Then H is a subspace of the vector space To... 

Readers who are still uncomfortable with thinking of functions as vectors should take the time to review this section carefully and do some exercises. These vector spaces are fundamental to our analysis of the hydrogen atom. In particular, we will look at the function space containing the wave functions for the hydrogen atom, and we will work with various subspaces of that function space. 

Exercise 2.13 Suppose V is a complex vector space of finite dimension. Suppose W is a subspace ofV and dim W = dim V. Show that W = V. 

Exercise 3.23 Show that C ([—1, 1]) is a complex vector space. Show that the set of complex-valued polynomials in one variable is a vector subspace. Show that the bracket ( , ) (defined as in Section 3.2) is a complex scalar product on C ([—1, 1]). 

Exercise 5.7 Recall the representations R of SU (2) on homogeneous polynomials introduced in Section 4.6. Find a complex scalar product on the vector space of the representation 7 i 7 2 such that the representation is unitary. Consider the subspace Vi spanned by uxy — vx, uy — rxy and the subspace Vj spanned by [ux", 2uxy + vx, 2vxy + uy, vy". Use this complex scalar product to find Is your answer isomorphic to V- Is it equal to V3 ... 

For some representations, the largest and smallest subspaces are the only invariant ones. Consider, for example, the natural representation of the group G = 50(3) on the three-dimensional vector space C . Suppose W is an invariant subspace with at least one nonzero element. We will show that W = C-. In other words, we will show that only itself (all) and the trivial subspace 0 (nothing) are invariant subspaces of this representation. It will suffice to show that the vector (1, 0, 0) lies in W, since W would then have to contain both... 

For the proof, it is helpful to recall the vector space Hom(Fi, Vfi), the vector space of linear transformations from Vi to V2, as well as the subspace Honif (V. V2) of homomorphisms of representations from Vi to lA- These were introduced in Section 5.5. 

Proposition 6.11 implies that irreducible representations are the identifiable basic building blocks of all finite-dimensional representations of compact groups. These results can be generalized to infinite-dimensional representations of compact groups. The main difficulty is not with the representation theory, but rather with linear operators on infinite-dimensional vector spaces. Readers interested in the mathematical details ( dense subspaces and so on) should consult a book on functional analysis, such as Reed and Simon [RS],... 

This Lie algebra is usually denoted gf ( , C) and is sometimes called the general linear (Lie) algebra over the complex numbers. Although this algebra is naturally a complex vector space, for our purposes we will think of it as a real Lie algebra, so that we can take real subspaces.We encourage the reader to check the three criteria for a Lie bracket (especially the Jacobi identity) by direct calculation. 

One assumption of the model is that each measurable quantity A (also known as an obser abJe ) of a finite-dimensional quantum system P( V) determines a decomposition of the vector space V into orthogonal subspaces, with a measurement value corresponding to each subspace. In other words, there... 

Finally we have the tools to state the answer to our original question what is the final state of a measured particle Consider a measurement A on a finite-dimensional vector space V, possibly with multiplicities. Suppose that a particle enters the measuring device in a state [u] and the measurement yields the result A. Let denote the subspace of states whose measurement is sine to yield A. Note that [u] because there is a nonzero chance... 

Next, we consider the symmetry operations of the system. The free energy is expanded as a function of the strains (as defined above) and the corresponding harmonic polynomials A (a,-). The resulting expression must be invariant under the symmetry transformations. If the symmetry is low enough, one can reduce further the vector space(s) introduced above, by choosing a suitable basis. The resulting irreducible subspaces are indicated... 

The following terminology is important The set ft = z,... xt of vectors x, 6 S is linearly dependent, iff there exists a set of scalars a,. ..at, not all zero, such that orixi + —h a = 0. If this is not possible, then the vectors are linearly independent. A vector x, for which a, 0 is one of the linearly dependent vectors. The set of vectors defines a vector subspace S, of S, called span(ft), which consists of all possible vectors z = aix, + —h atzt. This definition also provides a mapping from the array., a ) e Rk to the vector space span(ft). If ft is a linearly independent set, then the dimension of S, is k, and then the vectors constitutes a basis set in Si. If it is linearly dependent, then there is a subset fti 6 ft of size ki = card (ft,) which is linearly independent and spans the same space. Then ki is the dimension of S,. 

Proof. Consider the dual vector space 93s- This is isomorphic to an S dimensional vector space, 5ls, in which 2IT is embedded. Let b be the subspace of 93s which is isomorphic to the subspace of Us within which the set s/s lies. By Theorem 2 the dimension of b is t. Now the proper reactions of 93s are the subspace b of annihilators of b. By a well-known theorem on dual vector spaces [d], b must have dimension (s - t ). ... 

That is, the dimensionality of a vector space is the maximum number of LI vectors in that space. This is illustrated in Figure 3.2 for the example of two-dimensional (2-D) space, which is a subspace of 3-D space. 

These are nothing but the conditions of orthogonality of the subspace of interest lm/ (lm/ - image P - stands here for the set of vectors of a linear space which are obtained by action of the linear operator P upon all vectors of the linear vector space) and its complementary subspace IrriQ. 

We showed back in (3.4) that an algebraic G can be embedded in the general linear group of some vector space we now must refine that so that we can pick out a specified subgroup as the stabilizer of a subspace. Recall from (12.4) that f W is any subspace of some V where G acts linearly, the stabilizer Hw(R)= ge G R) g(W R) W R does form a closed subgroup. 

Definition 38 A linear subspace of L is a subset of L that forms a linear vector space under the rules of addition and scalar multiplication defined for L. 

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