Vector space linear transformation

The principal topics in linear algebra involve systems of linear equations, matrices, vector spaces, linear transformations, eigenvalues and eigenvectors, and least-squares problems. The calculations are routinely done on a computer. 

Here, we have introduced some rather abstract concepts (vector spaces, linear transformations) to analyze the properties of linear algebraic systems. For a fuller theoretical treatment of these concepts, and their extension to include systems of differential equations, consult Naylor Sell (1982). 

We have used for the row vectors of the respective entities, while we denote by ( ) and O the orbitals and many-electron functions, and by O and T(0) the two corresponding linear transformations, respectively. Various types of many-electron space for which such transformations may be carried out have been described by Malmqvist [34], In general, O may be non-unitary, possibly with subsidiary conditions imposed for ensuring that the corresponding transformation of the V-electron space exists e.g. a block-diagonal form according to orbital subsets or irreducible representations). 

Since the standard tableaux functions and the HLSP functions span the same vector space, a linear transformation between them is possible. Specifically, it would appear that the task is to determine the a,y s in... 

The mathematics we shall need is confined to the properties of vector spaces in which the scalar values are real numbers. From a mathematical viewpoint the whole discussion will take place in the context of two vector spaces, an S-dimensional space of chemical mechanisms and a Q-dimen-sional space of chemical reactions, which are related to each other by the fact that each mechanism m is associated with a unique reaction R(m) which it produces. The function R is a transformation of mechanisms to reactions which is linear by virtue of the fact that reactions are additive in a chemical system and that the reaction associated with combined mechanisms mt + m2 is R(m,) + R(m2). All mechanisms are combinations of a simplest kind of mechanism, called a step, which ideally consists of a one-step molecular interaction. Each step produces one of the elementary reactions which form a basis for the space of all reactions. 

Exercise 1.5 (Geometry of multiplication in C) 77zc complex plane can be considered as a two-dimensional real vector space, with basis 1, i. Show that multiplication by any complex number c is a linear transformation. Find the matrix for multiplication by i in the given basis. Find the matrix for multiplication by e, where d is a real number. Find the matrix for multiplication by a + ib. where a and b are real numbers. 

The notion of a linear transformation is crucial. A function from a (complex) vector space to a (complex) vector space is a (complex) linear transformation if it preserves addition and (complex) scalar multiplication. Here is a more explicit definition. 

Proposition 2.4 Suppose V is a finite-dimensional complex vector space and S is a subset ofV that spans V. Suppose W is a complex vector space. Suppose f S W is a function. Then there is a unique linear transformation T-. V -> VF such that for any s e S we have... 

Not only are linear transformations necessary for the very definition of a representation in Chapter 6, but they are useful in calculating dimensions of vector spaces — see Proposition 2.5. Linear transformations are at the heart of homomorphisms of representations and many other constructions. We will often appeal to the propositions in this section as we construct linear transformations. For example, we will use Proposition 2.4 in Section 5.3 to define the tensor product of representations. 

Isomorphisms are particularly important linear transformations because they tell us that domain and range are the same as far as vector space operations are concerned. 

Definition 2.7 Suppose V and W are vector spaces and T V —> W is a linear transformation. IfT is invertible and T W V is a linear transformation, then we say that T is an isomorphism of vector spaces (or isomor-phism/t r short) and that V and W are isomorphic vector spaces. 

On the other hand, many linear transformations are not isomorphisms. For one example, define a linear transformation from to the vector space P3 by... 

We will often want to consider linear transformations from a vector space... 

Proposition 2.10 Suppose V and W are finite-dimensional vector spaces and A V —> W and B.W V are linear transformations. Then... 

Exercise 2.7 Consider the complex plane C as a real vector space of dimension two. Is complex conjugation a real linear transformation ... 

Exercise 2.9 (Relevant to Proposition 7.3) Suppose V is the vector space of all polynomials in three variables. Suppose q is a polynomial in three variables. Show that multiplication by q is a linear transformation. In other words, consider the function taking any p(x, y,f)eV to qix, y, z) p x, y, z). Show that this function is linear. What is its range Remark these statements hold true for polynomials in any number of variables.) Now let denote the homogeneous polynomials in three variables of degree I. Let r denote the polynomial + y + z . Show that r" ... 

Exercise 2.14 (Used in Section 5.5) Let V denote a complex vector space. Let y denote the set of complex linear transformations from V to C. Show that y is a complex vector space. Show that ifV is finite dimensional then dim y = dim V. The vector space y is called the dual vector space or, more simply, the dual space. 

Exercise 2.21 Let denote the complex vector space of homogeneous complex-valued polyn omials of degree i in three real variables. Consider the linear transformation defined as the restriction of the Laplacian V" to P. Show that the image of this linear transformation lies in P ". 

Exercise 2.24 (Used in Exercise 3.20) Suppose V and W are vector spaces. DgyzneHom(V, IV) to be the set of linear transformations from V to W. Show that Honi(V, IV) is a vector space. Express its dimension in terms of the dimensions of V and W. 

Although our definition of adjoint applies only to finite-dimensional vector spaces, we cannot resist giving an inhnite-dimensional example. The proof of uniqueness works for infinite-dimensional spaces as well, but our proof of existence fails. Fix an element a e L (W) and consider the linear transformation T c defined by... 

Exercise 3.18 Any linear transformation T V —> V on a vector space V, satisfying = T is called a projection. Find a complex scalar product space V and a linear transformation T V —> V such that T is a projection but not an orthogonal projection. 

Definition 5.5 Suppose V is a complex vector space. The dual vector space of V, denoted V and pronounced "V -dual, is the complex vector space of linear transformations from V to C. 

The set of all linear transformations (not necessarily homomorptiisms of representations) from a representation V to a representation W forms a vector space too. This vector space is denoted Hom(T, IT). (Here Hom refers to the fact that a linear transformation can be considered a homomorphism of vector spaces.) There is a natural representation of G on this vector space. 

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. 

To calculate the dimension of the vector space H for every nonnegative integer f we will use the Fundamental Theorem of Linear Algebra (Proposition 2.5), which we repeat here if T is a linear transformation from a finitedimensional vector space V to a finite-dimensional vector space W, then we have... 

Proof. Consider the vector spaces of homogeneous polynomials of degree (. and P3 of homogeneous polynomials of degree f — 2 in three variables. (Sticklers for rigor should define Pf. = Pf .= 0. ) Let denote the restriction of the Laplacian = -f 9 -j- to P. By I xercise 2.21 we know that the image of the linear transformation Vf lies in P. ... 

Like the raising and lowering operators, the Casimir operator does not correspond to any particular element of the Lie algebra 5m(2). However, for any vector space V, both squaring and addition are well defined in the algebra gt (V) of linear transformations. Given a representation, we can define the Casimir element of that representation. ... 

Recall from Section 10.4 that if an observer undergoes a rotation of g (with g e 50(3), the spin-1/2 state space (C ) transforms via the linear operator Pi(g), while the spin-1 state space P(C ) transforms via pi(g). Hence the corresponding transformation of a vector v 0 w in 0 is... 

We will exploit the vector space isomorphism between the scalar product space Vo and the scalar product space Hom(V,, Vjf), introduced in Proposition 5.14 and Exercise 5.22. (Note that (V ) = V hy Exercise 2.15.) Instead of working directly with x 7 0, we will work with the corresponding linear transformation X 0. We will show that X Vq Etc has rank one and that its image is generated by an elementary element of the tensor product Ei 0 0 E . Then we will deduce that x itself is elementary in the tensor product Eo El 0 0 E . 

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