# Linear Algebra (Modular Mathematics Series)

Language: English

Pages: 240

ISBN: 0340610441

Format: PDF / Kindle (mobi) / ePub

As the basis of equations (and therefore problem-solving), linear algebra is the most widely taught sub-division of pure mathematics. Dr Allenby has used his experience of teaching linear algebra to write a lively book on the subject that includes historical information about the founders of the subject as well as giving a basic introduction to the mathematics undergraduate. The whole text has been written in a connected way with ideas introduced as they occur naturally. As with the other books in the series, there are many worked examples.

Solutions to the exercises are available online:

* The full set of solutions for lecturers only is available for download on a password protected website at http://textbooks.elsevier.com/

* readers are freely accessible at http://books.elsevier.com/companions/0340610441.

Geometry: A Metric Approach with Models (Undergraduate Texts in Mathematics)

Introduction to Real Analysis (Dover Books on Mathematics)

Infinity and the Mind: The Science and Philosophy of the Infinite (Princeton Science Library)

Linear Partial Differential Equations for Scientists and Engineers (4th Edition)

Real and Convex Analysis (Undergraduate Texts in Mathematics)

v = (a1 + ib1,…, an + ibn) is a vector with complex number entries, and if = (a1 - ib1,…, an - ibn), show that the dot product v. is a real number and that v. > 0 unless v = 0, the zero vector. 13. For vectors v1 = (a1, b1, c1), v2 = (a2, b2, c2) in 3 we define their vector product to be the vector v1 ∧ v2 = (b1c2 - b2c1, c1a2 - c2a1a1b2 - a2b1). Show that (i) v1 ∧ v2 = −(v2 ∧ v1), (ii) v1 ∧ (v1 ∧ v2) = v2 ∧(v1 ∧ v2) = (0, 0, 0) {so that v1 and v2 are each orthogonal to v1 ∧ v2} and find

the result to be another one. [In the first the product of two solutions to Euler’s equation is rarely another] In some spaces, however; the vectors themselves can be multiplied to produce new vectors. That this is true for the 3-dimensional physical/geometrical vectors emerged from the work of Hamilton and Grassmann (though the latter’s Ausdehnungslehre in 1844 was so impenetrable that it – and its 1862 successor – were largely ignored). Hamilton’s quaternions, resulting from his attempt to push

dimensional. (For, if it were spanned by polynomials p1(x), p2(x),…, pk(x), and if the greatest degree of these pi(x) were, say, t, then all polynomials in [x] would have to be of degree at most t, a blatant impossibility!) TUTORIAL PROBLEM 7.3 Is the trivial subspace {(0, 0,…, 0)} of n finite dimensional? One final remark. We have not yet defined the dimension of a finite dimensional vector space! What should the dimension of, say, 3 be? And, surely, subspaces of finite dimensional

itself, have smaller dimension than that of V. Furthermore n does indeed have dimension n. More important, the space of all solutions of the differential equation also has dimension n. For the m × n matrix A the row rank (column rank) of A is the dimension of the row space (column space) of A. These two ranks are equal so we call each the rank of A. The dimension of the subspace {x: x ∈ n and Ax = 0} (the so-called null space) of n is called the nullity of A. We then have: rank A + nullity A

multiplication, 96, 110 Codes and linear dependence, 136–7 and matrices, 48 Coefficient matrices, 20 Coefficients, in equations, 7, 10 Cofactors, definition, 76 Column rank, 149 theorem, 149–50 Column space, 115 Combination, linear, of vectors, 114 Commutative laws and matrices, 42 and vector spaces, 99 Completing the square, 205, 212 Complex numbers and eigenvalues/eigenvectors, 185 set of, 98 Complex vector space, 101 Component (≡ element≡ entry), of a matrix, 20 Conic