Author: haroonkhan

The possibility of expressing a vector as a linear combination of other vectors, or lack thereof, plays a role in many settings. In order to do so, we must ensure that the set of vectors that we want to use as a building blocks is “rich enough.” If we are given a set of vectors ,…

Consider a stylized economy with three possible future states of the world, as illustrated in Fig. 3.9. Say that three securities are available and traded on financial markets, with the following state-contingent payoffs: These vectors indicate, e.g., that asset 1 has a payoff 1 if state 1 occurs, a payoff 2 if state 2 occurs, and…

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…