Month: February 2023

We explore the connections between linear algebra and calculus. This is necessary in order to generalize calculus concepts to functions of several variables; since any interesting management problem involves multiple dimensions, this is a worthy task. The simplest nonlinear function of multiple variables is arguably a quadratic form: Denoting the double sum as  is typically preferred to ,…

In Section 3.4.3 we observed that a square matrix  is a way to represent a linear mapping from the space of n-dimensional vectors to itself. Such a transformation, in general, entails both a rotation and a change of vector length. If the matrix is orthogonal, then the mapping is just a rotation. It may happen, for a specific vector v and…

From a formal perspective, we may use matrix inversion to solve a system of linear equations: From a practical viewpoint, this is hardly advisable, as Gaussian elimination entails much less work. To see why, observe that one can find each column  of the inverse matrix by solving the following system of linear equations: Here, vector ej is a…

The determinant of a square matrix is a function mapping square matrices into real numbers, and it is an important theoretical tool in linear algebra. Actually, it was investigated before the introduction of the matrix concept. In Section 3.2.3 we have seen that determinants can be used to solve systems of linear equations by Cramer’s rule. Another…

In this section we explore the link between a basis of a linear space and the possibility of finding a unique solution of a system of linear equations Ax = b, where , , and . Here, n is the number of variables and m is the number of equations; in most cases, we have m = n, but we may try to generalize a bit. Recall that…

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…