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The purpose is to provide a text for the undergraduate linear algebra course of Massachusetts Institute of technology and was designed for mathematics majors at the junior level. In chapter 1, the author deals with systems of linear equations and their solutions by means of elementary row of operations on matrices. It provide sthe student with some picture of the origin of linear algebra and the computational techniques necessary to understand examples of more abstract ideas of the latter chapters. In Chapter 2, it talks with vector spaces, subspace, bases, and dimension. Chapter 3 treats linear transformations, their algebra, their representation by matrices, as well as isomorphism, linear functional and dual spaces. Chapter 4 prime factorization of a polynomial. It also deals with roots, tailor’s formula, at the defines the algebra of polynomials over a field, the ideals in that algebra, and the LaGrange interpolation formula. Chapter 5 develops determinants of square matrices, the determinant being viewed as an alternating n-liner function of the rows of a matrix, and then proceeds to multilinear functions on modules as well as the grassman ring. The materials molecules place the concepts of determinant in a wider and more comprehensive setting than a usually found in elementary textbooks. Chapter 6 and 7 contents a discussion of the concepts which are the basic of the analysis of a single linear transformation on a finite-dimensional vector space; the analysis and the characteristic (eigen) values, Triangulable and diagonalizable transformations; chapter 7 includes a discussion of matrices over a polynomial domain, the computation of invariant factors and elementary devisors of a matrix. The chapter ends with the discussion of Simi-simple operators, to round out the analysis of a single operator. Chapter 8 treats finite-dimensional inner product spaces in some detail. it Covers the basic geometry, relating orthogonalization to the idea of ‘best approximation to a vector’ and leading to a concepts of the orthogonal projection of a vector onto a subspace and orthogonal complement of a subspace. The chapter treats unitary operators and culminates in digitalization of self-adjoin and normal operators. Chapter 9 introduces sesqui-linear forms, relates them to positive and self-adjoin operators on an inner product space, move on to the spectral theory of normal operators and then to more sophisticated results concerning normal operators on a real or complex inner product spaces. Chapter 10 discusses bilinear forms, emphasizing canonical forms of symmetric and skew-symmetric forms, as well s group preserving non-degenerate forms, especially the orthogonal, unitary, pseudo-orthogonal and Lorentz groups.
- Linear Equations
- Vector space
- Linear transformations
- Polynomials
- Determinants
- Elementary Canonical forms
- The Rational and Jordan forms
- Inner products Spaces
- Operators in inner products spaces
- Bilinear forms
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