Dual space linear algebra pdf

In this course you will be expected to learn several things about vector spaces of course. Muhammad khalid of university of sargodha, sargodha written by atiq ur rehman. Please be aware, however, that the handbook might contain. I have seen a total of one proof of this claim, in jacobsons lectures in abstract algebra ii. Matrices and linear transformations 148 7a the matrix of a linear transformation 148 7b multiplication of transformations and matrices 153 7c the main theorem on linear transformations 157 7d rank and nullity of matrices 161 chapter 8. When fnis referred to as an inner product space, you should assume that the inner product. Recall that the dual space of a normed linear space x is the space of all bounded linear functionals from x to the scalar. It also deals with roots, taylors formula, and the lagrange inter polation. Linear algebra in physics summer semester, 2006 1 introduction the mathematical idea of a vector plays an important role in many areas of physics. Vector spaces, bases, linear maps, matrices and linear maps, direct sums, affine maps, the dual space, duality, gaussian elimination, lu, cholesky, echelon form, determinants, vector norms and matrix norms, eigenvectors and eigenvalues, iterative methods for solving linear systems, euclidean spaces.

Also, there is a unique linear functional on v, called the zero functional, which sends everything in v to zero. Today in lecture we discussed them and i understand what they are, but i dont really understand why we want to study them within. With this in mind, in figure 1 we have a diagram of how one might do this with v r2 and w x,y. The dual space as defined above is defined for all vector spaces, and to avoid ambiguity may also be called the algebraic dual space. One can, but in general should not, try to visualize the quotient space vw as a subspace of the space v. Dual spaces are useful in that they allow us to phrase many important concepts in linear algebra without the need to introduce. The material is developed completely from scratch, but at a faster pace than a beginning linear algebra course. The dual space first i remind you that a linear functional on a vector space v over r is any linear transformation f.

A very important theorem in linear algebra that is rarely taught is. The dual space of v, denoted by v, is the space of all linear functionals on v. Paper 1, section i 1f linear algebra state and prove the steinitz exchange lemma. These notes are for a graduate course in linear algebra. A vector space has the same dimension as its dual if and only if it is finite dimensional. Dual spaces friday 3 november 2005 lectures for part a of oxford fhs in mathematics and joint schools linear functionals and the dual space dual bases annihilators an example the second dual. The truth is, we will not so much use vector spaces in the study of linear systems as we will instead have linear systems start us on the study of vector spaces. Linear functionals, and the dual space once again let b. Pdf given any vector space v over a field f, the dual space v. Notes on dual spaces santiago canez in these notes we introduce the notion of a dual space. I also included new for 2009 chapter8, dealing with dual spaces and tensors. Sakai proved that any calgebra which is the dual of some banach space has a representation as a walgebra on some hilbert space. It is assumed that the reader has already studied matrix algebra or linear algebra, however, these notes are completely selfcontained.

All this gives the set of linear functionals the structure of a vector space. Vector spaces handwritten notes msc msc notes these are lecture notes of prof. Permutations and determinants 171 8a permutations 171 8b determinants 179. Thinking about a particle traveling through space, we imagine that its speed and direction of travel can be represented by a vector v in 3dimensional euclidean space r3. First i remind you that a linear functional on a vector space v over r is any linear transformation f. In mathematics, any vector space v has a corresponding dual vector space or just dual space for short consisting of all linear functionals on v, together with the vector space structure of pointwise addition and scalar multiplication by constants. A vector space v is a collection of objects with a vector. When defined for a topological vector space, there is a subspace of the dual space, corresponding to continuous linear functionals, called the continuous dual space. It is intended for a student who, while not yet very familiar with abstract reasoning, is willing to study more rigorous mathematics than what is presented in a \cookbook style calculus type course. When i first took linear algebra, we never learned about dual spaces. I feel that the material there, especially sections about tensors, is a bit too advanced for a rst year linear algebra course, but some topics for example, change of coordinates in the dual space can be easily included in the syllabus. Quotient spaces are emphasized and used in constructing the exterior and the symmetric algebras of a vector space.

For a vector space v over f, we define the dual space v homv,f as the set of. If w 0 62 im then the linear equation v w 0 has no solution, but we may. Dual spaces and linear functionals in this video, i introduce the concept of a dual space, which is the analog of a shadow world version, but for vector spaces. The most important example of an inner product space is fnwith the euclidean inner product given by part a of the last example. The following examples give some idea of the ubiquity of dual spaces in linear algebra. If you are a student and nd the level at which many of the current beginning linear algebra. Give an example to show that r 3 may be the direct sum of the kernel of and the image of, and also an example where this is not the case. Linear algebra notes lecture notes, university of toronto, fall 2016 1. Applications version 1 by howard anton and chris rorres and linear algebra and its applications 10 by gilbert strang are loaded with applications.

Vectors and spaces linear algebra math khan academy. Linear algebra notes lecture notes, university of toronto. Vector spaces definition part1 linear algebra duration. Dual spaces given a vector space v, one can consider the space of linear maps v. Linear algebra ii lecture notes pdf 61p download book. Chapter 4 defines the algebra of polynomials over a field, the ideals in that algebra, and the prime factorization of a polynomial. This handbook is intended to assist graduate students with qualifying examination preparation. Being examples of linear maps, we can add linear functionals and multiply them by scalars. In these notes we introduce the notion of a dual space. V f that assigns a number from field f to each vector x. With the dual space of l1 is meant l10, the space of bounded linear functionals of l1. You can prove it on almost the same way, as you proved that c0,1, with the k. The exterior algebra of a vector space and that of its dual are used in treating linear geometry. Note that v and v are isomorphic vector spaces, since they have the same dimension, but there is, in general, no speci c natural isomorphism between them and it is important to distinguish.

The wide variety of examples from this subsection shows that the study of vector spaces is interesting and important in its own right, aside from how it helps us understand linear systems. It supposed to be a rst linear algebra course for mathematically advanced students. Linear algebradefinition and examples of vector spaces. Dual spaces are useful in that they allow us to phrase many important concepts in linear algebra without. The dual space and the dual map 101 the null space and range of the dual of a linear map 104 the matrix of the dual of a linear map 109 the rank of a matrix 111 exercises 3. Interesting to note is that the dual space x0, of a normed space x, is a banach space. F 1 4 polynomials 117 complex conjugate and absolute value 118 uniqueness of coef cients for polynomials 120 the division algorithm for polynomials 121 zeros of. First of all, together with tensor products, dual spaces can be used to talk about linear transformations. Chapter 11 linear functionals and the dual space 349 11. Lets get our feet wet by thinking in terms of vectors and spaces. Sheldon axler linear algebra done right yikun zhang. An introductory overview of vector spaces, algebras, and linear geometries over an arbitrary commutative.