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1、Review of basic concepts and facts in linear algebra,Matrix Analysis HITSZ Instructor: Zijun Luo Fall 2012,guideline,Prerequisite: Text: Matrix Analysis, Roger A. Horn, Charles R. Johnson, Cambridge University Press, Reprint edition (February 23,1990). ISBN 0521386322, 575p Homework assignment: week
2、ly Exam: final exam (open book) on Jan. 14, 2013 (temporary) Content: vector space, norms, eigenvalues, unitary matrix, Hermitian matrix, matrix factorization, canonical form, nonnegative matrix,Definition : Vector Space ( V, + ; F ) A vector space (over a field F) consists of a set V along with 2 o
3、perations + and s.t. For the vector addition + : v, w, u V v + w V ( Closure ) v + w = w + v ( Commutativity ) ( v + w ) + u = v + ( w + u ) ( Associativity ) 0 V s.t. v + 0 = v ( Zero element ) v V s.t. v v = 0 ( Inverse ) (2) For the scalar multiplication : v, w V and a, b F, a v V ( Closure ) ( a
4、 + b ) v = a v + b v ( Distributivity ) a ( v + w ) = a v + a w ( a b ) v = a ( b v ) = a b v ( Associativity ) 1 v = v (Scalar identity of multiplication),Expression of force, velocity, gradient,Subspace A set U is a subspace of a vector space V if Every element of U is in V, and U is a vector spac
5、e.,Linear Combinations,Consider a set of vectors v1,.,vn and a set of scalars a1, , an A linear combination of the vectors is a1v1+a2v2+anvn,Remark: Vector space = Collection of linear combinations of vectors.,Definition : Span Let S = s1 , , sn | sk ( V,+,R ) be a set of n vectors in vector space V
6、. The span of S is the set of all linear combinations of the vectors in S, i.e.,with,Lemma : The span of any subset of a vector space is a subspace.,Proof:,Let S = s1 , , sn | sk ( V,+,R ) ,and,QED,Note: span S is the smallest vector space containing all members of S.,Example:,Proof: The problem is
7、tantamount to showing that for all x, y R, unique a,b R s.t.,i.e.,has a unique solution for arbitrary x & y.,Since,QED,Example: 1+x , 1x is linearly independent.,Proof:,Let,Otherwise they are linearly independent.,Example:,Let,then S = v1 ,v2 , v3 is L.D. Note: v3-2v2=0,Basis,Definition : Basis A ba
8、sis of a vector space V is an ordered set of linearly independent (non-zero) vectors that spans V, i.e. any vector in V can be represented as a linear combination of the basis.,Example 1.2:,is a basis for R2,B is L.I. :,B spans R2:,Present an example on board,Distance: make sense,Basic concepts: vec
9、tor space, subspace, span, linear combination, linearly independent, linear dependent, basis, dimension, vector norm, inner product.,Important principles: *A span is a subspace. *Zero vector is l.d. to all vectors; *Subset of a l.i. set is l.i.; *L.i. vectors can be added to form a basis; *Every basis has the same number of vectors; *Each vector has a unique basis-representation; *Every inner product has the Cauchy-Schwarz inequality. * Every inner product can be used to define a vector norm; * Every vector norm is a continuous function; *All vector norms are equivalent;,CONCLUSION,HOMEWORK,
