Algebra DeMYSTiFieD (2nd Edition) by Rhonda Huettenmueller PDF

By Rhonda Huettenmueller

ISBN-10: 0071743626

ISBN-13: 9780071743624

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A square matrix has two diagonals. The one leading from the upper left corner to the lower right corner is called the main diagonal or simply the diagonal and the other one the secondary diagonal. A square matrix is called diagonal if all its entries outside of the (main) diagonal are zero. Multiplication by a diagonal matrix looks especially simple: a1 0 ... 0 b11 0 a2 ... 0 b21 o o ... an b1 b12 ... b22 ... b1p b2 bnP = aibll a1b12 ... a2b21 a2b22 ... anbnl anbn2 ... albla a2b2p I (every row of the second matrix is multiplied by the respective diagonal entry of the first matrix).

Vector Spaces In elementary geometry, we not only add vectors but also multiply them by numbers. Analyzing properties of these two operations, we come to the notion of a vector space. 24 1. Algebraic Structures Before actually defining it, we should note that here we step away from what we previously meant by an operation. Multiplication of a vector by a number is not an operation on two elements of the same set. It is an operation that assigns to each pair (number, vector) another vector. This is how things also are in the general definition of a vector space; however, there elements of an arbitrary (fixed) field replace real numbers.

The linearly independent system {el, e2i... , eh l can be completed to a basis of V. Thus, if U # V, dim V > k. 41. Determine the number of k-dimensional subspaces of an n-dimensional vector space over a field of q elements. The next theorem provides a complete description of all finite-dimensional vector spaces. 42. Finite-dimensional vector spaces over the same field are isomorphic if and only if their dimensions are the same. Proof. If f : V - U is an isomorphism of vector spaces and {el, e2, ...

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