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Unformatted text preview: Math 210 Jerry L. Kazdan Vectors and an Application to Least Squares This brief review of vectors assumes you have seen the basic properties of vectors previously. We can write a point in R n as X = ( x 1 ,..., x n ) . This point is often called a vector . Fre quently it is useful to think of it as an arrow pointing from the origin to the point. Thus, in the plane R 2 , X = ( 1 , 2 ) can be thought of as an arrow from the origin to the point ( 1 , 2 ) . Algebraic Properties Alg1. ADDITION: If Y = ( y 1 ,..., y n ) , then X + Y = ( x 1 + y 1 ,..., x n + y n ) . Example : In R 4 , ( 1 , 2 , 2 , )+( 1 , 2 , 3 , 4 ) = ( , 4 , 1 , 4 ) . Alg2. MULTIPLICATION BY A CONSTANT: cX = ( cx 1 ,..., cx n ) . Example : In R 4 , if X = ( 1 , 2 , 2 , ) , then 3 X = ( 3 , 6 , 6 , ) . Alg3. DISTRIBUTIVE PROPERTY: c ( X + Y ) = cX + cY . This is obvious if one writes it out using components. For instance, in R 2 : c ( X + Y ) = c ( x 1 + y 1 , x 2 + y 2 ) = ( cx 1 + cy 1 , cx 2 + cy 2 ) = ( cx 1 , cx 2 )+( cy 1 , cy 2 ) = cX + cY . Length and Inner Product IP1. k X k : = q x 2 1 + + x 2 n is the distance from X to the origin. We will also refer to k X k as the length or norm of X . Similarly k X Y k is the distance between X and Y . Note that k X k = 0 if and only if X = 0, and also that for any constant c we have k cX k =  c k X k . Thus, k 2 X k = k 2 X k = 2 k X k . IP2. The inner product of X and Y is, by definition, h X , Y i : = x 1 y 1 + x 2 y 2 + + x n y n . (1) (this is also called the dot product and written X Y ). The inner product of two vectors is a number, not another vector. In particular, we have the vital identity k X k 2 = h X , X i . Example : In R 4 , if X = ( 1 , 2 , 2 , ) and Y = ( 1 , 2 , 3 , 4 ) , then h X , Y i = ( 1 )( 1 ) + ( 2 )( 2 )+( 2 )( 3 )+( )( 4 ) = 3. IP3. ALGEBRAIC PROPERTIES OF THE INNER PRODUCT. These are obvious from the above definition of h X , Y i . h X + Y , W i = h X , W i + h Y , W i , h cX , Y i = c h X , Y i , h Y , X i = h X , Y i . 1 REMARK: If one works with vectors having complex numbers as elements, then the definition of the inner product must be modified to h X , Y i : = x 1 y 1 + x 2 y 2 + + x n y n , (2) where y 1 means the complex conjugate of y 1 (note: many people put the complex conjugate on the x j instead of the y j ). The purpose is to insure that the fundamental property k X k 2 = h X , X i still holds. Note, however, that the property h Y , X i = h X , Y i is now replaced by h Y , X i = h X , Y i . IP4. GEOMETRIC INTERPRETATION: h X , Y i = k X kk Y k cos , where is the angle between X and Y . Since cos ( ) = cos , the sense in which we measure the angle does not matter....
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This note was uploaded on 03/06/2012 for the course MATH 260 taught by Professor Staff during the Spring '12 term at UPenn.
 Spring '12
 STAFF
 Vectors, Least Squares

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