University:
Whitman College
Course:
MATH-244 | Differential Equations
Academic year:

2024
Views:

233

Pages:

6
Author:

absolument20yzj

Day 2 of Linear Systems (Replaces 7.2) • What is a matrix? Determinant? Trace? • Be able to convert a matrix-vector equation into a system of equations and vice versa. • Be able to write the solution to the homogeneous system of equations in vector form (also- Equilibrium solutions) • Recall how to write lines in 2d, 3d using vector format. • Understand the vector ekt v, especially geometrically. • If ansatz if x(t) = eλt v, what must be true for x0 = Ax? Class Notes First some useful notation. Key Definition: A system of equations can be written in matrix-vector form as shown below (this is a definition) ax + by = e a b x e ⇔ = cx + dy = f c d y f Similarly, we could extend this to three variables (this is look like):  a11 a12 a11 x1 + a12 x2 + a13 x3 = b1  a21 a22 a21 x1 + a22 x2 + a23 x3 = b2 ⇔ a31 a32 a31 x1 + a32 x2 + a33 x3 = b3 just to show you what it would     b1 x1 a13 a23   x2  =  b2  b3 x3 a33 More Definitions... A matrix is simply an array of numbers, and the size of a matrix is defined as the number of rows × the number of columns (similar to a spreadsheet, rows always come first). We identify elements of the array by locating the row and column. In the examples above, we have a 2 × 2 matrix and a 3 × 3 matrix. A vector can also be described as a 2 × 1 matrix (or 3 × 1). The definitions above gives meaning to matrix-vector multiplication. For example, 1 2 1 1(1) + 2(2) 5 = = −1 3 2 −1(1) + 3(2) 5 1 In the previous lesson, we reviewed the determinant. The trace of a matrix is the sum of the diagonal elements of a matrix (from upper left element to the lower right element). For example, 1 2 A= det(A) = 4 − 6 = −2 Tr(A) = 1 + 4 = 5 3 4 Review: Solving a System of Equations In solving a linear system of equations, there are three (and only three) possible outcomes: (i) Exactly one solution (intersecting lines), (ii) No Solution (parallel lines), (iii) an infinite number of solutions (the same line). We can solve the system using Cramer’s Rule: ax + by = e cx + dy = f ⇒ e f a c x= b d b d y= a c a c e f b d This works as long as the determinant in the denominator is not zero. If that is the case, then we either have parallel lines or the same line. Example Solve the system: 1 2 2 4 x y = 0 0 SOLUTION: The determinant is 0, and we see these are the same line. Therefore, any (x, y) satisfying x + 2y = 0 will solve the system- We want to represent the line in vector form. x −b Shortcut: The line ax + by = 0 can be represented by the vector =t y a x −2 To solve the previous system, we should provide the line: =t y 1 Lines in Vector Form Recall from Calculus III: A line in two or three dimensions can be defined if it goes through point p and goes in the direction q as (in parametric vector form): p~ + t~q −∞

Day 2 of Linear Systems - Class Notes

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