Mathematics_1_oneside.pdf

Problems 72 verify that a system of linear equations

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— Problems 7.2 Verify that a system of linear equations can indeed written in ma- trix form. Moreover show that each equation Ax = b represents a system of linear equations. 7.3 Prove Lemma 7.3 and Theorem 7.4 . 7.4 Let A = a 0 1 . . . a 0 m be an m × n matrix. (1) Define matrix T i j that switches rows a 0 i and a 0 j . (2) Define matrix T i ( α ) that multiplies row a 0 i by α . (3) Define matrix T i j ( α ) that adds row a 0 j multiplied by α to row a 0 i . For each of these matrices argue why these are invertible and state their respective inverse matrices. H INT : Use the results from Exercise 4.14 to construct these matrices. 7.5 Prove Lemma 7.8 . 7.6 Prove Theorem 7.9 . Use a so called constructive proof. In this case this means to pro- vide an algorithm that transforms every input matrix A into row reduce echelon form by means of elementary row operations. De- scribe such an algorithm (in words or pseudo-code).
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8 Euclidean Space We need a ruler and a protractor . 8.1 Inner Product, Norm, and Metric Inner product. Let x , y R n . Then Definition 8.1 x 0 y = n X i = 1 x i y i is called the inner product ( dot product , scalar product ) of x and y . Fundamental properties of inner products. Let x , y , z R n and α , β Theorem 8.2 R . Then the following holds: (1) x 0 y = y 0 x (Symmetry) (2) x 0 x 0 where equality holds if and only if x = 0 (Positive-definiteness) (3) ( α x + β y ) 0 z = α x 0 z + β y 0 z (Linearity) P ROOF . See Problem 8.1 . In our notation the inner product of two vectors x and y is just the usual matrix multiplication of the row vector x 0 with the column vector y . However, the formal transposition of the first vector x is often omitted in the notation of the inner product. Thus one simply writes x · y . Hence the name dot product. This is reflected in many computer algebra sys- tems like Maxima where the symbol for matrix multiplication is used to multiply two (column) vectors. Inner product space. The notion of an inner product can be general- Definition 8.3 ized. Let V be some vector space. Then any function ⟨· , ·⟩ : V × V R that satisfies the properties 55
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8.1 I NNER P RODUCT , N ORM , AND M ETRIC 56 (i) x , y ⟩ = ⟨ y , x , (ii) x , x ⟩ ≥ 0 where equality holds if and only if x = 0, (iii) α x + β y , z ⟩ = α x , z ⟩+ β y , z , is called an inner product . A vector space that is equipped with such an inner product is called an inner product space . In pure mathematics the symbol x , y is often used to denote the (abstract) inner product of two vectors x , y V . Let L be the vector space of all random variables X on some given prob- Example 8.4 ability space with finite variance V ( X ). Then map ⟨· , ·⟩ : L × L R , ( X , Y ) 7→ ⟨ X , Y ⟩ = E ( XY ) is an inner product in L . Euclidean norm. Let x R n . Then Definition 8.5 k x k = p x 0 x = s n X i = 1 x 2 i is called the Euclidean norm (or norm for short) of x .
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