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Unformatted text preview: EMCH 201: Introduction to Numerical Methods Linear Algebra A. Motivation: 1. Why are we doing this? i. Numerous times in engineering, one is required to solve a set of simultaneous linear equations. ii. We need an abbreviated way to represent and solve these types of equations. 2. Examples: i. Statics : In the plane you will have 3 linear equations with 3 unknowns. Later, you will increase this to 3 n equations in 3 n unknowns for trusses. ii. Robotics : When one tries to solve the equations between the speed of the joints and the speed of the robotic hand (aka gripper or end effector), it results in 6 equations with 6 unknowns. In other words, you are given the xyz speed and the roll pitch yaw rates of the end effector and you want to find the joint speeds. iii. Finite Element Analysis : In finite element analysis, one typically breaks up an object into small elements which one can write the governing equations for. In the end one can end up with thousands of equations with thousands of unknowns. B. The Basics: 1. Definitions: i. Linear Systems of Equations: A set of equations in which at least one unknown appears in several of the equations. This condition means that all the equations must be solved simultaneously. The system is linear if the unknowns only appear linearly, in other words, only multiplied by a constant or added (+/-) to other unknowns. These systems can be solved by rewriting the system of equations as a single matrix equation. The study of matrix arithmetic and how matrix equations are solved comes under the heading of Linear Algebra. Here is a system of n equations with n unknowns: n n nn n n n n n n n n n n n b x a x a x a x a x a x a b x a x a x a x a x a x a b x a x a x a x a x a x a b x a x a x a x a x a x a = + + + + + + = + + + + + + = + + + + + + = + + + + + + 5 5 4 4 3 3 2 2 1 1 3 3 5 35 4 34 3 33 2 32 1 31 2 2 5 25 4 24 3 23 2 22 1 21 1 1 5 15 4 14 3 13 2 12 1 11 ii. Matrix: A matrix is a two dimensional array. It has elements, and each is labeled by its row and column number, or position in the array. (sometimes called a 2-D array). The matrix A, below, is designated by its elements a ij , where the first number indicates the row and the second number the column. Singular; matrix, plural; matrices. = 44 43 42 41 34 33 32 31 24 23 22 21 14 13 12 11 a a a a a a a a a a a a a a a a A They are written within brackets, braces or parentheses, they are NOT written between bars or angular braces. Thus one can write: 4 3 2 1 = 4 3 2 1 = 4 3 2 1 But NOT 4 3 2 1 4 3 2 1 4 3 2 1 A matrix is described by its dimension which is its rows by columns. The first matrix above has dimension n x n , the second one is 4 x 4 and the ones immediately above are 2 x 2....
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- Fall '09