00-use - 9/20/2004, USE OF LINEAR ALGEBRA I Math 21b, O....

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Unformatted text preview: 9/20/2004, USE OF LINEAR ALGEBRA I Math 21b, O. Knill This is not a list of topics covered in the course. It is rather a lose selection of subjects for which linear algebra is useful or relevant. The aim is to convince you that it is worth learning this subject. Most of this handout does not make much sense yet to you because the objects are not defined yet. You can look at this page at the end of the course again, when some of the content will become more interesting. 1 2 3 4 GRAPHS, NETWORKS. Linear al- gebra can be used to understand networks . A network is a collec- tion of nodes connected by edges and are also called graphs. The ad- jacency matrix of a graph is de- fined by an array of numbers. One defines A ij = 1 if there is an edge from node i to node j in the graph. Otherwise the entry is zero. A prob- lem using such matrices appeared on a blackboard at MIT in the movie Good will hunting. How does the array of numbers help to understand the network. One ap- plication is that one can read off the number of n-step walks in the graph which start at the vertex i and end at the vertex j. It is given by A n ij , where A n is the n-th power of the matrix A . You will learn to compute with matrices as with numbers. CHEMISTRY, MECHANICS Complicated objects like a bridge (the picture shows Storrow Drive connection bridge which is part of the big dig), or a molecule (i.e. a protein) can be modeled by finitely many parts (bridge elements or atoms) coupled with attractive and repelling forces. The vibrations of the system are described by a differential equation x = Ax , where x ( t ) is a vector which depends on time. Differential equations are an important part of this course. The solution x ( t ) = exp( At ) of the differential equation x = Ax can be understood and computed by find- ing the eigenvalues of the matrix A. Knowing these frequencies is impor- tant for the design of a mechani- cal object because the engineer can damp dangerous frequencies. In chemistry or medicine, the knowl- edge of the vibration resonances al- lows to determine the shape of a molecule. QUANTUM COMPUTING A quantum computer is a quantum mechanical system which is used to perform computations. The state x of a machine is no more a sequence of bits like in a classical computer but a sequence of qubits , where each qubit is a vector. The memory of the computer can be represented as a vector. Each computation step is a multiplication x 7 Ax with a suitable matrix A . Theoretically, quantum computa- tions could speed up conventional computations significantly. They could be used for example for cryp- tological purposes. Freely available quantum computer language (QCL) interpreters can simulate quantum computers with an arbitrary num- ber of qubits....
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This note was uploaded on 04/06/2008 for the course MATH 21B taught by Professor Judson during the Spring '03 term at Harvard.

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00-use - 9/20/2004, USE OF LINEAR ALGEBRA I Math 21b, O....

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