Category: Linear Algebra

In the previous sections, we introduced vectors and matrices and defined an algebra to work on them. Now we try to gain a deeper understanding by taking a more abstract view, introducing linear spaces. To prepare for that, let us emphasize a few relevant concepts: Linear algebra is the study of linear mappings between linear…

In this section, we summarize a few useful properties of the matrix operations we have introduced. Some have been pointed out along the way; some are trivial to check, and some would require a technical proof that we prefer to avoid. A few properties of matrix addition and multiplication that are formally identical to properties…

Consider a matrix . When we multiply a vector  by this matrix, we get a vector . This suggests that a matrix is more than just an arrangement of numbers, but it can be regarded as an operator mapping  to : Given the rules of matrix algebra, it is easy to see that this mapping is linear, in the sense…

Matrix inversion is an operation that has no counterpart in the vector case, and it deserves its own section. In the scalar case, when we consider standard multiplication, we observe that there is a “neutral” element for that operation, the number 1. This is a neutral element in the sense that for any , we have x ·…

Operations on matrices are defined much along the lines used for vectors. In the following, we will denote a generic element of a matrix by a lowercase letter with two subscripts; the first one refers to the row, and the second one refers to the column. So, element aij is the element in row i, column j. Addition Just like…

The solution of systems of linear equations. Many issues related to systems of linear equations can be addressed by introducing a new concept, the matrix. Matrix theory plays a fundamental role in quite a few mathematical and statistical methods that are relevant for management. We have introduced vectors as one-dimensional arrangement of numbers. A matrix is, in…

The two basic operations on vectors, addition and multiplication by a scalar, can be combined at wish, resulting in a vector; this is called a linear combination of vectors. The linear combination of vectors vj with coefficients αj, Fig. 3.8 Illustrating linear combinations of vectors. j = 1, …, m is If we denote each component i, i = l, …, n, of vector j by vij, the component i of the linear…

The inner product is an intuitive geometric concept that is easily introduced for vectors, and it can be used to define a vector norm. A vector norm is a function mapping a vector x into a nonnegative number  that can be interpreted as vector length. We have see that we may use the dot product to define the usual Euclidean norm: It…

We are quite used to elementary operations on numbers, such as addition, multiplication, and division. Not all of them can be sensibly extended to the vector case. Still, we will find some of them quite useful, namely: Vector addition Addition is defined for pairs of vectors having the same dimension. If , we define: For instance Since…

Vectors are an intuitive concept that we get acquainted with in highschool mathematics. In ordinary two- and three-dimensional geometry, we deal with points on the plane or in the space. Such points are associated with coordinates in a Cartesian reference system. Coordinates may be depicted as vectors, as shown in Fig. 3.4; in physics, vectors are…