8.13 Affine combinations of solutions of linear equations. Consider the set of m linear equations in n variables Ax = b , where A is an m × n matrix, b is an m -vector, and x is the n -vector of variables. Suppose that the n -vectors z 1 , . . . , z k are solutions of this set of equations, i.e. , satisfy Az i = b . Show that if the coefficients α 1 , . . . , α k satisfy α 1 + · · · + α k = 1, then the affine combination w = α 1 z 1 + · · · + α k z k is a solution of the linear equations, i.e. , satisfies Aw = b . In words: Any affine combina- tion of solutions of a set of linear equations is also a solution of the equations. 8.14 Stoichiometry and equilibrium reaction rates. We consider a system (such as a single cell) containing m metabolites (chemical species), with n reactions among the metabolites occurring at rates given by the n -vector r . (A negative reaction rate means the reaction runs in reverse.) Each reaction consumes some metabolites and produces others, in known rates proportional to the reaction rate. This is specified in the m × n stoichiometry matrix S , where S ij is the rate of metabolite i production by reaction j , running at rate one. (When S ij is negative, it means that when reaction j runs at rate one, metabolite i is consumed.) The system is said to be in equilibrium if the total production rate of each metabolite, due to all the reactions, is zero. This means that for each metabolite, the total production rate balances the total consumption rate, so the total quantities of the metabolites in the system do not change. Express the condition that the system is in equilibrium as a set of linear equations in the reaction rates.
162 8 Linear equations
Chapter 9 Linear dynamical systems In this chapter we consider a useful application of matrix-vector multiplication, which is used to describe many systems or phenomena that change or evolve over time. 9.1 Linear dynamical systems Suppose x 1 , x 2 , . . . is a sequence of n -vectors. The index (subscript) denotes time or period, and is written as t ; x t , the value of the sequence at time (or period) t , is called the state at time t . We can think of x t as a vector that changes over time, i.e. , one that changes dynamically. In this context, the sequence x 1 , x 2 , . . . is sometimes called a trajectory or state trajectory . We sometimes refer to x t as the current state of the system (implicitly assuming the current time is t ), and x t +1 as the next state , x t - 1 as the previous state , and so on. The state x t can represent a portfolio that changes daily, or the positions and velocities of the parts of a mechanical system, or the quarterly activity of an econ- omy. If x t represents a portfolio that changes daily, ( x 5 ) 3 is the amount of asset 3 held in the portfolio on (trading) day 5. A linear dynamical system is a simple model for the sequence, in which each x t +1 is a linear function of x t : x t +1 = A t x t , t = 1 , 2 , . . . . (9.1) Here the n × n matrices A t are called the dynamics matrices . The equation above is called the dynamics or update equation, since it gives us the next value of x , i.e. , x t +1 , as a function of the current value
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