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Lec1 - MAT1341A LECTURE NOTES FOR FALL 2008 MONICA...

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MAT1341A: LECTURE NOTES FOR FALL 2008 MONICA NEVINS (HEAVILY BORROWING FROM HANDWRITTEN NOTES OF BARRY JESSUP) 2. Vector Geometry, Part I The material for this class and the next comes largely from [N, 3.1,3.2,3.3,3.5], but we also incorporate the generalizations to higher dimensions where appropriate. We will not cover everything in these sections, and we will add some material that is not present in the book. You are responsible for what we cover in class, the suggested exercises, what is on the homework assignments, and what is presented in the DGD. It is assumed that [N, 3.1,3.2,3.3] are substantially review for students having taken the prerequisites for this course; but this review gives us the opportunity to set the stage for the upcoming chapters. You are encouraged to look through those sections of the textbook to refresh your memory. One good quick technique is to consider all the words in boldface , and read more about any of those keywords which are unfamiliar. 2.1. Vectors in R n [N, 3.1] . Vector comes from the Latin vehere , which means to carry, or to convey; abstractly we think of the vector as taking us along the arrow that we represent it with. For example, we use vectors in Physics to indicate the magnitude and direction of a force. Let’s use our understanding of the geometry and algebra of vectors in low dimensions (2 and 3) to develop some ideas about the geometry and algebra in higher dimensions. “Algebra” “Geometry” R , real numbers, “scalars” real line R 2 = { ( x, y ) | x, y R } , vectors u = ( x, y ) plane R 3 = { ( x, y, z ) | x, y, z R } , vectors u = ( x, y, z ) 3-space . . . (why not keep going?) . . . R 4 = { ( x 1 , x 2 , x 3 , x 4 ) | x i R } , vectors x = ( x 1 , x 2 , x 3 , x 4 ) Hamilton (1843): extended C to “hamiltonians” “space-time” n Z , n > 0: R n = { ( x 1 , x 2 , · · · , x n ) | x i R } x = ( x 1 , · · · , x n ) n -space” [N, 4.1.2] Date : September 9, 2008. 1
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2 MONICA NEVINS (HEAVILY BORROWING FROM HANDWRITTEN NOTES OF BARRY JESSUP) Notation: We have several different notations which we use for writing vectors, which we use interchangeably: x = (1 , 2 , 3 , 4) x = 1 2 3 4 x = 1 2 3 4 T , where the “upper T” stands for “tranpose.” (“Transposition” means turning rows into columns (or vice versa) [N, 1.1.4].) The vertical notation (“matrix form”) is the easiest to read but the first one is easier to write; the book uses the last one a lot (also called matrix form). (The reason for the vertical vector notation is in [N, 1.4], which we’ll get to later.) 2.2. Manipulation of vectors in R n . The algebraic rules for R n extend directly from the algebraic rules for R 2 and R 3 : Equality: ( x 1 , · · · , x n ) = ( y 1 , · · · , y n ) x i = y i for all i ∈ { 1 , · · · , n } . (In particular, ( x 1 , · · · , x n ) = ( y 1 , · · · , y m ) if n = m .) Addition: ( x 1 , · · · , x n ) + ( y 1 , · · · , y n ) = ( x 1 + y 1 , · · · , x n + y n ) Zero vector: 0 = (0 , 0 , · · · , 0) R n Negative: if x = ( x 1 , · · · , x n ) then - x = ( - x
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