LinearAlgebra - University of Central Florida School of...

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Unformatted text preview: University of Central Florida School of Electrical Engineering and Computer Science EGN-3420 - Engineering Analysis. Fall 2009 - dcm Linear Algebra Concepts Vector Spaces To define the concept of a vector space we first need to introduce two basic algebraic structures, the group and the field. A group is a set G with one binary operation “ · ”, called multiplication, which satisfies three conditions 1. Associative law: ∀ ( a,b,c ) ∈ G a · ( b · c ) = ( a · b ) · c. 2. Identity element: There is an identity element e ∈ G such that a · e = e · a = a, ∀ a ∈ G . 3. Inverse element: ∀ a ∈ G, ∃ a- 1 such that a · a- 1 = a- 1 · a = e. A group G whose operation satisfies the commutative law (i.e., a · b = b · a ) is a commutative , or Abelian , group. A field is a set F equipped with two binary operations, addition and multiplication, with the following properties: 1. under addition, F is an Abelian group with the identity (or neutral) element 0 such that 0 + a = a, ∀ a ∈ F . 2. under multiplication, the nonzero elements form an Abelian group with neutral element 1 such that 1 · a = a, ∀ a ∈ F , and 0 · a = 0 , ∀ a ∈ F . The additive and multiplicative identity elements are different, 0 6 = 1. 3. the distributive law holds: a · ( b + c ) = a · b + a · c . A vector space A assumes three objects: 1. An Abelian group ( V, +) whose elements are called “vectors” and whose binary operation “+” is called addition , 2. A field F (usually R , the real numbers, or C , the complex numbers), whose elements are called “scalars”, and 3. An operation called “multiplication with scalars” and denoted by “ · ”, which associates to any scalar c ∈ F and vector α ∈ V a new vector c · α ∈ V . The multiplication by scalars operation has the following properties c · ( α + β ) = c · α + c · β ( c + c ) · α = c · α + c · α ( c · c ) · α = c · ( c · α ) , 1 · α = α where α,β ∈ V and c,c ∈ F. Observations: (a) Often we omit the “ · ” symbol and write the product of two scalars as cc instead of c · c and the product of a scalar with a vector as cα instead of c · α . (b) Given n scalars { c 1 ,c 2 ,...,c n } ∈ R , then the set of n vectors { α 1 ,α 2 ,...,α n } ∈ R n are linearly independent if c 1 α 1 + c 2 α 2 + ... + c n α n = 0 = ⇒ c 1 = c 2 = ... = c n = 0 . Vectors that are not linearly independent are called linearly dependent . A subspace S of a vector space A is a subset of A which is closed with respect to the operations of addition and scalar multiplication. This means that the sum of two vectors in S is in S . For any vector and any scalar, the product of a vector with a scalar is also a vector, ∀ α ∈ S and ∀ c ∈ R then cα ∈ S ....
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LinearAlgebra - University of Central Florida School of...

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