Matrices and Linear Transformations: Second Edition by Charles G. Cullen

By Charles G. Cullen

Undergraduate-level advent to linear algebra and matrix concept offers with matrices and linear structures, vector areas, determinants, linear changes, similarity, polynomials, and polynomial matrices. additionally spectral decomposition, Jordan canonical shape, answer of the matrix equation AX=XB, and over 375 difficulties, many with solutions. "Comprehensive." — digital Engineer's layout journal.

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Additional resources for Matrices and Linear Transformations: Second Edition

Sample text

5. Is {0, 1, 2, 3, 4} a field with + and · defined by the tables below? You may assume that the associative and distributive laws hold. 6. Let be any field. 1 show that a) b) c) d) 7. Show that the system (x) (see Example 5) is indeed a field. Be sure to consider what “ +,” “ ·,” and “ = ” mean in this system. 3 MATRICES The remainder of this chapter is concerned principally with matrices and their relationship to systems of linear algebraic equations. The first order of business is to formally define the term matrix.

5. Is {0, 1, 2, 3, 4} a field with + and · defined by the tables below? You may assume that the associative and distributive laws hold. 6. Let be any field. 1 show that a) b) c) d) 7. Show that the system (x) (see Example 5) is indeed a field. Be sure to consider what “ +,” “ ·,” and “ = ” mean in this system. 3 MATRICES The remainder of this chapter is concerned principally with matrices and their relationship to systems of linear algebraic equations. The first order of business is to formally define the term matrix.

The definition of the transpose of a matrix makes perfectly good sense independent of the nature of the elements of the matrix. For matrices whose elements are complex numbers, there is a related idea which turns out to be more useful in practice. If z = a + bi is a complex number, then the complex conjugate of z is = a – bi It is assumed that the reader has already had some experience with complex arithmetic, but for easy reference the relevant properties of the complex conjugate are listed below.