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33 eigenvalues and eigenvectors

# 33 eigenvalues and eigenvectors - 18.03 Lecture#33 Nov 23...

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18.03 Lecture #33 Nov. 23, 2009: notes The actual second half of the double lecture on Monday was mostly a continuation of the very general discussion of systems of ordinary differential equations, and was (therefore) included with the notes labelled Lecture 32. What I will do here is say the few things I actually said in lecture about linear systems, and then try to add (what I did not cover adequately in lecture) an introduction to matrices, eigenvalues, and eigenvectors. The vector space R n consists of the column vectors x = x 1 x 2 . . . x n , ( x j R , j = 1 , . . . , n ) . Writing the coordinates of x in a column instead of a row has some notational advantages, which should become very clear to you in 18.06; for now it may just look like a strange convention. There are two fundamental things we want to do with vectors: addition (of two vectors) and scalar multiplication (of a real number by a vector). These are defined by x + y = x 1 + y 1 x 2 + y 2 . . . x n + y n , c · x = c · x 1 c · x 2 . . . c · x n , ( x , y R n , c R ) . There are lots of notations used for vectors. Mathematicians very often write them with (italic, not bold) Roman letters, especially v and w . Especially on blackboards, where different typefaces are hard to distinguish, they are sometimes written with a little arrow, as vectorx , and I’ll often do that on the board. This notation shouldn’t really have survived in print, but it has. I’ll try to use x fairly consistently, but it isn’t really possible. Matrices (coming up soon!) are just a special kind of vector, but they are never written with bold type; capital (italic) Roman letters are fairly common, with the same lower case letter used for the coordinates (which are always called entries in a matrix. See how inconsistent we are?) In the world of real-valued functions of one variable x , a linear function is one of the form y ( x ) = ax + b , with a and b real constants. (Of course this is the equation of a straight line with slope a and y -intercept b .) In the world of vector-valued functions of vector variables, the word linear almost always excludes the constant b . Here’s the definition.

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