Linear Algebra Day 1

Linear Algebra Day 1 - ICME Refresher Course Lecture#1...

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ICME Refresher Course Lecture #1 Milinda Lakkam Institute for Computational and Mathematical Engineering September 19, 2011 Milinda Lakkam ICME Refresher Course 1 / 20
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Scalars, and Vectors, and Matrices. scalar: single quantity or measurement. (Greek) α,β,γ C . vector: ordered collection of scalars. (Lowercase) a,b,c . a = α 1 α 2 . . . α n C n All vectors are column vectors, unless otherwise specified. Denote row vectors as a T = ± α 1 ··· α n ² . matrix: two-dimensional collection of scalars. (Uppercase) A,B, Σ , Λ . A C m × n is a matrix with m rows and n columns A = ± a 1 ··· a n ² A = a T 1 . . . a T m Milinda Lakkam ICME Refresher Course 2 / 20
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Operations on vectors scalar multiplication: αv , α C . vector addition: c = a + b , when a C n and b C n . inner product / dot product / scalar product : h x,y i = n k =1 x k ¯ y k , when x C n ,y C n . Milinda Lakkam ICME Refresher Course 3 / 20
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Vector Norms Vector Norm is a mapping f : C n R 0 satisfying the following properties: 1 Scale invariant: f ( αx ) = | α | f ( x ) , α C and x C n 2 f ( x ) = 0 ⇐⇒ x = 0 C n 3 Triangle Inequality: f ( x + y ) f ( x ) + f ( y ) A norm f ( x ) is often denoted for convenience by || x || . Milinda Lakkam ICME Refresher Course 4 / 20
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Commonly used vector norms 1 The 2-Norm : f ( x ) = k x k 2 = v u u t n X i =1 ν 2 i . 2 The 1-Norm : f ( x ) = k x k 1 = n X i =1 | ν i | . 3 The -Norm : f ( x ) = k x k = max 1 i n | ν i | . This gives the component with largest magnitude. In all cases, the norm is a measure of length; the 2-Norm corresponds to the Euclidean length of the vector. Milinda Lakkam ICME Refresher Course 5 / 20
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1 Let a = { α i : 1 i n } and b = { β i : 1 i n } be elements in R n . We say a b if and only if α i β i for all 1 i n. 2
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This note was uploaded on 10/01/2011 for the course EE 221 taught by Professor Ee221a during the Spring '08 term at Berkeley.

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Linear Algebra Day 1 - ICME Refresher Course Lecture#1...

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