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set0 - $ Set 0 Review of Basics Kyle A Gallivan Department...

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a39 a38 a36 a37 Set 0: Review of Basics Kyle A. Gallivan Department of Mathematics Florida State University Numerical Linear Algebra Fall 2010 1
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a39 a38 a36 a37 Scalars, Vectors and Matrices Scalars and their operations are assumed to be from the field of real numbers ( R ) the field of complex numbers ( C ) complex number: α = β + where i here is used to represent the root of 1 (occasionally we will use j for this but it will be made clear when this is done) β and γ are the real and imaginary parts of α respectively complex conjugate ¯ α = β the absolute value of α denoted | α | is α ¯ α = p β 2 + γ 2 2
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a39 a38 a36 a37 Scalars, Vectors and Matrices R n – a vector is an one-dimensionally ordered list of n real scalars addition of vectors is componentwise scalar addition scalar vector product multiplies each component of the vector with the scalar C n – a vector is an one-dimensionally ordered list of n complex scalars addition of vectors is componentwise complex scalar addition scalar vector product multiplies each complex component of the vector with the complex scalar 3
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a39 a38 a36 a37 Example – R 3 Vectors: x = 0 B B @ 1 3 52 1 C C A y = 0 B B @ 10 4 2 1 C C A Basic Operations: x + y = 0 B B @ 11 1 50 1 C C A 2 x = 0 B B @ 2 6 104 1 C C A 3 y = 0 B B @ 30 12 6 1 C C A Linear Combination: 2 x + 3 y = 0 B B @ 32 6 98 1 C C A 4
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a39 a38 a36 a37 Example – R 3 e 1 = 0 B B @ 1 0 0 1 C C A e 2 = 0 B B @ 0 1 0 1 C C A e 3 = 0 B B @ 0 0 1 1 C C A e = 0 B B @ 1 1 1 1 C C A 5
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a39 a38 a36 a37 Scalars, Vectors and Matrices Definition 0.1. An m × n matrix of scalars from R or C is a two-dimensionally ordered arrangement of mn scalars A = 0 B B B B B B @ α 11 α 12 · · · α 1 n α 21 α 22 · · · α 2 n . . . . . . . . . α m 1 α m 2 · · · α mn 1 C C C C C C A The set of m × n matrices with scalar elements from R is denoted R m × n The set of m × n matrices with scalar elements from C is denoted C m × n 6
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a39 a38 a36 a37 Matrix Operations Matrix scaling A, B R m × n and γ R : B = γA = has elements β ij = γα ij Matrix addition A, B, C R m × n : C = A + B = B + A has elements γ ij = β ij + α ij This is the collection of vectors R mn and the associated scalar field and operations 7
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a39 a38 a36 a37 Matrix Vector Product Definition 0.2. If A = a 1 a 2 · · · a n R m × n and the vector x R n x = 0 B B B B B B @ ξ 1 ξ 2 . . . ξ n 1 C C C C C C A then Ax = a 1 ξ 1 + a 2 ξ 2 + · · · + a n ξ n 8
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a39 a38 a36 a37 Matrix Operations If A R n 1 × n 2 , B R n 2 × n 3 , then C R n 1 × n 3 is Scalar definintion: C = AB has elements γ ij = Σ n 2 k =1 α ik β kj Matrix-vector definition: C = AB c i = Ab i i = 1 , . . . , n 3 where c i = Ce i , b i = Be i Outer product definition: C = AB = Σ n 2 i =1 a i b T i where a i = Ae i , b T i = e T i B Inner product definintion: C = AB has elements γ ij = a T i b j where b i = Be i , a T i = e T i A 9
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a39 a38 a36 a37 Matrix Operations the matrix product is not commutative the matrix product is associative the matrix product is distributive, i.e., A ( B + C ) = AB + AC All scalars and vectors can be replaced with submatrices of appropriate dimension to yield block forms of the matrix product 10
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