Matrix Theory -vector space School of Mathematical Sciences Teaching Group Main Reference books Fuzhen Zhang.Matrix Theory-Basic Results and Techniques,Second Edition. Springer,2011. llse C.F.Ipsen,Numerical Matrix Analysis:Linear Systems and Least Squares. SIAM,2009. Reference books: Roger A.Horn and Charles A.Johnson:Matrix Analysis.Cambridge University Press,1985. Gene H.Golub and Charles F.Van Loan:Matrix Computations,Third Edition. Johns Hopkins Press,1996. Nicholas J.Higham.Accuracy and Stability of Numerical Algorithms,Second Edition.SIAM,2002. Y.Saad.Iterative Methods for Sparse Linear Systems,Second Edition.SIAM, Philadelphia,2003. Matrix Theory Matrices Maintained by Yan-Fei Jing1
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Elementary Linear Algebra Review Vector Spaces Let V be a set of objects(elements)and F be a field,mostly the real number field R or the complex number field C throughout this course The set V is called a vector space over F if the operations addition u+v,u,v∈V and scalar multiplication cw,c∈F,v∈V, are defined so that the addition is associative,is commutative,has an additive identity 0 and additive inverse-v in V for each vV. and so that the scalar multiplication is distributive,is associative, and has an identity 1EF for which 10=o for every vV. 2
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l.u+v∈V for all u,v∈V. 2.cuw∈V for all e∈F and v∈V. 3.u+v=v+u for all u,vEV. 4.(u+v)+w=u+(v+w)for all u,v,w E V. 5.There is an element 0 V such that v+0=v for all vV. 6.For each vV there is an element-v V so that v+(-v)=0. 7.c(u+v)=cu+cu for all c∈F and u,v∈V. 8.(a+b)v=aw+bu for all a,b∈F and v∈V. 9.(ab)v=a(bw)for all a,b∈andv∈V. 10.1v=v for all v∈V. U u+U c,c>1 1 Vector addition and scalar multiplication 3
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We call the elements of a vector space vectors and the elements of the field scalars.For instance,R",the set of real column vectors 1 also written as (1,x2,....n)T (r for transpose)is a vector space over R with respect to the addition (c1,2,,xn)T+(1,2,,n)T-(c1+1,2十2，，n十n)T and the scalar multiplication c (1,22,....In)T (cx1;cx2;...,cun)T,cR. 4
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Note that the real row vectors also form a vector space over R; and they are essentially the same as the column vectors as far as vector spaces are concerned.For convenience,we may also consider R"as a row vector space if no confusion is caused.However,in the matrix-vector product Ax,obviously r needs to be a column vector. Let S be a nonempty subset of a vector space V over a field F. Denote by Span S the collection of all finite linear combinations of the vectors in S;that is,Span S consists of all vectors of the form C1v1+C22+…+Ctt,t=1,2,,c∈F,吃∈S, The set Span S is also a vector space over F.If Span S=V,then every vector in V can be expressed as a linear combination of vectors in S.In such cases we say that the set S spans the vector space V. A set S={v1,v2,...,vk}is said to be linearly independent if C1U1+C2U2+·+CkVk=0 holds only when ci =c2=...=ck=0.If there are also nontrivial solutions,i.e.,not all c are zero,then S is linearly dependent. 5
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For example,both{(1,0),(0,1),(1,1)}and{(1,0),(0,1)}span R2.The first set is linearly dependent;the second one is linearly independent.The vectors(1,0)and (1,1)also span R2. 6
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A basis of a vector space V is a linearly independent set that spansV. If V possesses a basis of an n-vector set S={v1,v2,...,Un}, V is of dimension n,written as dimV =n. V={0,we write dimV =0. For instance, C is a vector space of dimension 2 over R with basis {1,i,where i=√-I, C is a vector space of dimension 1 over C with basis {1. 7
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C",the set of row (or column)vectors of n complex components, is a vector space over C standard basis e1=(1,0,,0,0),e2=(0,1,.,0,0),,en=(0,0,.,0,1) 8
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If {u1,u2,...,un}is a basis for a vector space V of dimension n, then every x in V can be uniquely expressed as a linear combination of the basis vectors: T=x1l1+x22十··+Cnun, where the zi are scalars. The n-tuple (1,22,...,n)is called the coordinate of vector x with respect to the basis. 9
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Let V be a vector space of dimension n,and let {v1,v2,...,vk}be a linearly independent subset of V. if k <n,then there exists a vector v+1E V such that the set fv1,v2,...,Uk,v+1 is linearly independent. It follows that the set fv1.v2.....v can be extended to a basis of V. 10
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