数据库文化基础 (5).pdf

Linear Independence3rdweek/Linear AlgebraObjectives of This Week2The goal is to understandLinear independenceUniqueness of a solution in a linear systemSubspace,basis,and dimensionColumn space and rank 3 Recall the matrix equation of a linear system:Or,a vector equation is written as Recall:Linear System6065551+5.55.06.02+1013=66747811+22+33=Person IDWeightHeightIs_smokingLife-span160kg5.5ftYes(=1)66265kg5.0ftNo(=0)74355kg6.0ftYes(=1)78605.51655.00556.01123=667478x =x =4 The solution exists only when Span 1,2,3.If the solution exists for =,when is it unique?It is unique when 1,2,and 3are linearly independent.Infinitely many solutions exist when 1,2,and 3are linearly dependent.Uniqueness of Solution for =6065551+5.55.06.02+1013=66747811+22+33=5(Practical)Definition:Given a set of vectors v v1,v v,check if v vcan be represented as a linear combination of the previous vectors v v1,v v2,v v1for =1,e.g.,v v Span v v1,v v2,v v1for some =1,?If at least one such v vis found,then v v1,v v is linearly dependent.If no such v vis found,then v v1,v v is linearly independent.Linear Independence6(Formal)Definition:Consider 1v v1+2v v2+v v=.Obviously,one solution is =12=000,which we call a trivial solution.v v1,v vare linearly independent if this is the only solution.v v1,v vare linearly dependent if this system also has other nontrivial solutions,e.g.,at least one being nonzero.Linear Independence7 If v v1,v vare linearly dependent,consider a nontrivial solution.In the solution,lets denote as the last index such that 0.Then,one can write=11 11,and safely divide it by,resulting in v v=1v v1 1v v1 Span v v1,v v2,v v1which means can be represented as a linear combination of the previous vectors.Two Definitions are Equivalent8 Given two vectors 1and 2,Suppose Span 1,2 is the plane on the right.When is the third vector 3linearly dependent of 1and 2?That is,v v3 Span v v1,v v2?Geometric Understanding of Linear Dependencev v22v v23v v2v v12v v1x x=2v v1+3v v29 A linearly dependent vector does not increase Span!If v v3 Span v v1,v v2,then Span 1,2=Span 1,2,3,Why?Suppose 3=1v v1+2v v2,then the linear combination of 1,2,3can be written as 1v v1+2v v2+3v v3=1+1v v1+1+1v v2which is also a linear combination of v v1,v v2.Linear Dependence10 Also,a linearly dependent set produces multiple possible linear combinations of a given vector.Given a vector equation 1v v1+2v v2+3v v3=,supposethe solution is =123=321,i.e.,3v v1+2v v2+1v v3=.Suppose also 3=2v v1+3v v2,a linearly dependent case.Then,3v v1+2v v2+1v v3=3v v1+2v v2+2v v1+3v v2=5v v1+5v v2,so =123=550is another solution.Many more solutions exist.Linear Dependence and Linear System Solution11 Actually,many more solutions exist.e.g.,3v v1+2v v2+1v v3=3v v1+2v v2+2v v3 1v v3=3v v1+2v v2+2 2v v1+3v v2 1v v3=7v v1+8v v2 1v v3,thus =123=781is another solution.Linear Dependence and Linear System Solution12 The solution exists only when Span 1,2,3.If the solution exists for =,when is it unique?It is unique when 1,2,and 3are linearly independent.Infinitely many solutions exist when 1,2,and 3are linearly dependent.Uniqueness of Solution for =6065551+5.55.06.02+1013=66747811+22+33=13 Definition:A subspace is defined as a subset of closed under linear combination:For any two vectors,1,2,and any two scalars and,1+2.Span v v1,v v is always a subspace.Why?1=1v v1+v v,2=1v v1+v v 1+2=1v v1+v v+1v v1+v v=1+1v v1+v v In fact,a subspace is always represented as Span v v1,v v.Span and Subspace14 Definition:A basis of a subspace is a set of vectors that satisfies both of the following:Fully spans the given subspace Linearly independent(i.e.,no redundancy)In the previous example,where =Span 1,2,3,Span 1,2 forms a plane,but 3=2v v1+3v v2 Span 1,2,1,2 is a basis of,but not 1,2,3 nor 1 is a basis.Basis of a Subspace15 Consider a subspace (green plane).Is a basis unique?That is,is there any other set of linearly independent vectors that span the same subspace?Non-Uniqueness of Basisv v22v v23v v2v v12v v1x x=2v v1+3v v216 What is then unique,given a particular subspace?Even though different bases exist for,the number of vectors in any basis for will be unique.We call this number as the dimension of,denoted as dim.In the previous example,the dimension of the plane is 2,meaning any basis for this subspace contains exactly two vectors.Dimension of Subspace17 Definition:The column space of a matrix is the subspace spanned by the columns of.We call the column space of as Col.A=111001Col =Span 110,101 What is dim Col?Column Space of Matrix Given A=112101011,note that 211=110+101,i.e.,the third column is a linear combination of the first two.What is dim Col?Col =Span 110,101,211Col =Span 110,10118Matrix with Linearly Dependent Columns19 Definition:The rank of a matrix,denoted by rank,is the dimension of the column space of i.e.,rank =dim Col Rank of MatrixLinear independence Uniqueness of a solution of a linear systemSubspace,basis,and dimension,column space,and rankSummary