Lets get our feet wet by thinking in terms of vectors and spaces. Wewillcallu a subspace of v if u is closed under vector addition, scalar multiplication and satis. Systems of linear equations are rephrased in terms of matrix equations, i. Lecture 9 sum and intersection of two subspaces, examples. These linear algebra lecture notes are designed to be presented as twenty ve, fty minute lectures suitable for sophomores likely to use the material for applications but still requiring a solid foundation in this fundamental branch of mathematics. The help that it gives to studentscomesfromtakingadevelopmentalapproachthisbookspresentation emphasizesmotivationandnaturalness,usingmanyexamples. Number systems and fields, vector spaces, linear independence, spanning and bases of vector spaces, subspaces, linear transformations, matrices, linear transformations and matrices, elementary operations and the rank of a matrix, the inverse of a linear. These eight conditions are required of every vector space.
In this monograph we make the standing assumption that all vector spaces use either the real or the complex numbers as scalars, and we say real vector spaces and complex vector spaces to specify whether real or complex numbers are being used. Examples of vector spaces vector subspaces span, linear dependence, linear independence systems of linear equations bases overview of course this course is an introduction to linear algebra. First, construct two copies of the real lineone horizontal and one verticalwhich intersect perpendicularly at their origins. Vector spaces are the basic setting in which linear algebra happens. In reality, linear algebra is the study of vector spaces and the functions of vector spaces linear transformations. If youre behind a web filter, please make sure that the domains. The notion of scaling is addressed by the mathematical object called a. I had trouble understanding abstract vector spaces when i took linear algebra i hope these help. Linear algebra as an introduction to abstract mathematics. Introduction to vector spaces this section introduces the concept of vector space. Linear algebra ii lecture notes pdf 61p download book. Finite dimensional vector space, linear dependent and independent, related theorem.
Math 2040 matrix theory and linear algebra ii 4 chapter 4 lecture notes. Pdf lecture 11 rank of a matrix, solvability of system of linear. A powerful result, called the subspace theorem see chapter 9 guarantees, based on the closure properties alone, that homogeneous solution sets are vector spaces. That leads to linear geometry and the language of vectors and vector spaces. Column space and nullspace in this lecture we continue to study subspaces, particularly the column space and nullspace of a matrix. Vector spaces come in many disguises, sometimes containing objects which do not at all look like vectors. Linear transformation, ranknullity theorem, row and column space. For any vector space v we have slv, the special linear lie algebra of v, consisting of the operators on v of trace 0. Vector spaces and linear transformations beifang chen fall 2006 1 vector spaces a vector space is a nonempty set v, whose objects are called vectors, equipped with two operations, called addition and scalar multiplication. Linear algebra lecture notes martin bright and daan krammer pdf 56p this book explains the following topics related to linear algebra.
Lecture notes on linear algebra department of mathematics. Topics in our linear algebra and tensor analysis notes pdf in these linear algebra and tensor analysis notes pdf, you will study the concept of generalized mathematical constructs in terms of algebraic structures mainly vector spaces and tensors. This course offers a rigorous treatment of linear algebra, including vector spaces, systems of linear equations, bases, linear independence, matrices, determinants, eigenvalues, inner products, quadratic forms, and canonical forms of matrices. A vector space v is a collection of objects with a vector. The lectures were distributed to the students before class, then posted on a. Among the dozens and dozens of linear algebra books that have appeared, two that were written before \dumbing down of textbooks became fashionable are especially notable, in my opinion, for the clarity of their authors mathematical vision. Pdf lecture 7 span, linearly independent, basis, examples. Such vectors belong to the foundation vector space rn of all vector spaces.
These lecture notes correspond to the course linear algebra ii, as given at. They form the fundamental objects which we will be studying throughout the remaining course. Vector spaces a vector space is an abstract set of objects that can be added together and scaled according to a speci. Every vector space has a unique zero vector satisfying 0cv dv. Linear algebra is the mathematics of vector spaces and their subspaces. These linear algebra lecture notes are designed to be presented as twenty five. In general, all ten vector space axioms must be veri. Lecture notes on linear algebra by david lerner department of mathematics university of kansas and the students of math 291 fall, 2007 these are notes of a course given in fall, 2007 to the honors section of our elementary linear algebra course. Surprisingly, many of these \unexpected vector spaces play a. Linear algebra in twenty five lectures uc davis mathematics. However, if w is part of a larget set v that is already known to be a vector space, then certain axioms need not. All vector spaces have to obey the eight reasonable rules.
The main idea of the course is to emphasize the concepts of vector spaces and linear transformations as mathematical structures that can be used to model the world around us. More generally, if \v\ is any vector space, then any hyperplane through the origin of \v\ is a vector space. It is used by the pure mathematician and by the mathematically trained scientists of all disciplines. A supplementary notes on matrices and linear systems 164. A vector space over a eld f consists of a set v the elements of which are called vectors along with an addition operation. These linear algebra lecture notes are designed to be presented as twenty ve, fty minute lectures suitable for sophomores likely to use the material for. Paul halmoss finitedimensional vector spaces 6 and ho man and kunzes linear algebra 8. Those are three of the eight conditions listed in the chapter 5 notes. Notes on linear algebra school of mathematical sciences queen.
Alan doolittle basis vectors a minimal set of orthogonal vectors that uniquely and. Lecture notes for math 115a linear algebra terence tao. Linear algebra is foundational for mathematics and has applications in many parts of physics, including classical mechanics, electromagnetism, quantum mechanics, general relativity etc. We will elaborate on all of this in later chapters, but let us demonstrate the main features of a 1 2.
Note that c is also a vector space over r though a di erent one from the previous example. To find a basis for the latter, we write down the equations for the system. The aim of this course is to provide an introduction to the study of nite dimensional vector spaces over elds of characteristic zero and linear morphisms between them and to provide an abstract understanding of several. Then, given a pair x 1, x 2, the first coordinate, x 1, specifies the points. A the characteristic polynomial of aas a function of tra the trace of the matrix a 5. Direct sums of vector spaces thursday 3 november 2005 lectures for part a of oxford fhs in mathematics and joint schools direct sums of vector spaces projection operators idempotent transformations two theorems direct sums and partitions of the identity important note. The symbols fxjpxg mean the set of x such that x has the property p. Linear algebra is one of the most applicable areas of mathematics. Abstract vector spaces, linear transformations, and their coordinate representations contents 1 vector spaces 1. The null space of a matrix a is equal to the null space of its reduced row echelon form r. In this section, we introduce the \arena for linear algebra.
The product of any scalar c with any vector u of v exists and is a unique vector of. Vector space axioms page 3 definition of the scalar product axioms in a vector space, the scalar product, or scalar multiplication operation, usually denoted by, must satisfy the following axioms. Vectors and spaces linear algebra math khan academy. A vector space is a nonempty set v of objects, called vectors, on which are defined two operations, called addition and multiplication by scalars real numbers. We will see that many questions about vector spaces can be reformulated as questions about arrays of numbers. Abstract vector spaces, linear transformations, and their. Vector spaces these notes are about linear maps and bilinear forms on vector spaces, how we represent them by matrices, how we manipulate them, and what we use this for. Martin bright and daan krammer warwick, january 2011.
The translation is straightforward but after mastering the basics of \matrix arithmetic one must interpret those basics in geometric terms. The definition of the vector operations is usually obvious. Each such pair specifies a point in the plane as follows. On the theoretical side, we deal with vector spaces, linear maps, and bilin. Lecture notes for math 115a linear algebra fall of 2002 terence tao, ucla.
These lecture notes correspond to the course linear algebra ii. Why we need vector spaces by now in your education, youve learned to solve problems like the one. These notes are about linear maps and bilinear forms on vector spaces, how we. There are vectors other than column vectors, and there are vector spaces other than rn. If youre seeing this message, it means were having trouble loading external resources on our website. Vector spaces and linear maps artificial intelligence.
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