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vectors6

# vectors6 - Math 210 Jerry L Kazdan Vectors and an...

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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 Alg-1. 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 , 0 )+( - 1 , 2 , 3 , 4 ) = ( 0 , 4 , 1 , 4 ) . Alg-2. MULTIPLICATION BY A CONSTANT : cX = ( cx 1 ,..., cx n ) . Example : In R 4 , if X = ( 1 , 2 , - 2 , 0 ) , then - 3 X = ( - 3 , - 6 , 6 , 0 ) . Alg-3. 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 IP-1. X : = x 2 1 + ··· + x 2 n is the distance from X to the origin. We will also refer to X as the length or norm of X . Similarly X - Y is the distance between X and Y . Note that X = 0 if and only if X = 0, and also that for any constant c we have cX = | c | X . Thus, - 2 X = 2 X = 2 X . IP-2. The inner product of X and Y is, by definition, X , Y : = 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 X 2 = X , X . Example : In R 4 , if X = ( 1 , 2 , - 2 , 0 ) and Y = ( - 1 , 2 , 3 , 4 ) , then X , Y = ( 1 )( - 1 ) + ( 2 )( 2 )+( - 2 )( 3 )+( 0 )( 4 ) = - 3. IP-3. ALGEBRAIC PROPERTIES OF THE INNER PRODUCT . These are obvious from the above definition of X , Y . X + Y , W = X , W + Y , W , cX , Y = c X , Y , Y , X = X , Y . 1

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REMARK : If one works with vectors having complex numbers as elements, then the definition of the inner product must be modified to X , Y : = 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 X 2 = X , X still holds. Note, however, that the property Y , X = X , Y is now replaced by Y , X = X , Y . IP-4. GEOMETRIC INTERPRETATION : X , Y = X Y cos θ , where θ is the angle between X and Y . Since cos ( - θ ) = cos θ , the sense in which we measure the angle does not matter. To see this, we can restrict our attention to the two dimensional plane containing X and Y . Thus, we need consider only vectors in R 2 . Assume we are not in the trivial case where X or Y are zero. Let α and β be the angles that X = ( x 1 , x 2 ) and Y = ( y 1 , y 2 ) make with the horizontal axis, so θ = β - α . Then x 1 = X cos α and x 2 = Y sin α . Similarly, y 1 = Y cos β and y 2 = Y sin β . Therefore X , Y = x 1 y 1 + x 2 y 2 = X Y ( cos α cos β + sin α sin β ) = X Y cos ( β - α ) = X Y cos θ . This is what we wanted. IP-5. GEOMETRIC CONSEQUENCE : X and Y are perpendicular if and only if X , Y = 0, since this means the angle θ between them is 90 degrees so cos θ = 0. We often use the word orthogonal as a synonym for perpendicular .
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vectors6 - Math 210 Jerry L Kazdan Vectors and an...

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