# lec4 - McGill University Math 270 Applied Linear Algebra...

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Unformatted text preview: McGill University Math 270: Applied Linear Algebra CHAPTER 4: INNER PRODUCT SPACES AND THE GRAM-SCHMIDT ORTHOGONALIZATION PROCEDURE 1 Introduction In this section, we are going to extend the the idea of dot prod- uct for geometric vectors to a general vector space. With the dot product, one can define length and orthogonality in the vec- tor space. Recall that for two geometric vectors in R 3 space, x = ( x 1 ,x 2 ,x 3 ) y = ( y 1 ,y 2 ,y 3 ) , one has the inner product x · y = k x kk y k cos θ = ( x 1 y 1 + x 2 y 2 + x 3 y 3 ) . Based on this inner product, one can introduce the length of the vector, k x k 2 = x · x = x 2 1 + x 2 2 + x 2 3 . 0-0 and further the angle between two vectors: cos θ = x · y k x kk y k . Thus, one may define two vectors x , y orthogonal, if and only if x · y = 0 . Now for a general vector space, by defining the inner prod- uct, we can also introduce the geometric concepts of length or norm, orthogonality. We start with the vector space R n . 1.1 Definition of Inner Product in R n Definition 1.1.1 Given two vectors in R n , x = ( x 1 ,x 2 , ··· ,x n ) , y = ( y 1 ,y 2 , ··· ,y n ) . We define the standard inner product , h x , y i , in R n by h x , y i = x · y = x 1 y 1 + x 2 y 2 + ··· + x n y n . (1) The norm of x is k x k = p h x , x i = q x 2 1 + x 2 2 + ··· + x 2 n . (2) Due to k x k ≥ for any x ∈ V , we can prove that 0-1 1.1.1 Schwarz Inequality |h x , y i| = | x · y | ≤ k x kk y k . or | x · y | k x kk y k ≤ 1 . Thus, it is possible to define the angle θ between two vec- tors: cos θ = x · y k x kk y k . Proof: As for any number α , k x + α y k = ( x + α y ) · ( x + α y ) ≥ , we have F ( α ) = k x k 2 + 2 α h x , y i + α 2 k y k 2 = aα 2 + 2 bα + c ≥ , F ( α ) is a quadratic function of α , which cannot have two real roots. Therefore, we deduce that b 2- ac ≤ , namely h x , y i 2-k x k 2 k y k 2 ≤ . 0-2 1.1.2 Triangle Inequality k x + y k ≤ k x k + k y k . Proof: k x + y k 2 = ( x + y ) · ( x + y ) = x · x + 2 x · y + y · y ≤ k x k 2 + 2 k x kk y k + k y k 2 ≤ ( k x k + k y k ) 2 1.2 Basic Properties of the Inner Product in R n It is seen that for all vectors, x , y , z in R n , and all real number c , we have 1. h x , x i ≥ , and h x , x i = 0 , if and only if x = 0 . 2. h x , y i = h y , x i . 3. h c x , y i = c h x , y i . 4. h x + y , z i = h x , z i + h y , z i . 0-3 2 Definition of a Real Inner Product Space Definition 2.0.1 ( inner product ) Let V be a real vector space. An inner product in V is a mapping that associates a real number with any pair of vectors x , y ∈ V , denoted by h x , y i = x · y , and satisfies the following properties: For all vectors, x , y , z in V , and all real number c , 1. h x , x i ≥ , and h x , x i = 0 , if and only if x = 0 ....
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lec4 - McGill University Math 270 Applied Linear Algebra...

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