数据库文化基础 (4).pdf

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1、Linear Combinations2ndweek/Linear AlgebraObjectives of This Week2The goal is to understandLinear combination and vector equationSpanExistence of solutions in a linear systemMatrix-matrix multiplication as linear combination of vectors3 Given vectors v v1,v v2,v vin and given scalars 1,2,is called a

2、linear combination of v v1,v vwith weights or coefficients 1,.The weights in a linear combination can be any real numbers,including zero.Linear Combinations1v v1+v v4 Recall the matrix equation of a linear system:A matrix equation can be converted into a vector equation:From Matrix Equation to Vecto

3、r Equation6065551+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 =5 Consider its vector equation:When does the solution exist for =?Existence of Solution for =6065551+5.55.06.02+1013=6674

4、7811+22+33=6 Definition:Given a set of vectors v v1,v v,Span v v1,v v is defined as the set of all linear combinations of v v1,v v.That is,Span v v1,v v is the collection of all vectors that can be written in the formwith arbitrary scalars 1,.Span v v1,v v is also called the subset of spanned(or gen

5、erated)by v v1,v v.Span1v v1+2v v2+v v7 If 1are 2nonzero vectors in 3,with 2not a multiple of 1,then Span 1,2 is the plane in 3that contains 1,2and 0 0.In particular,Span 1,2 contains the line in 3through 1and 0 0 and the line through 2and 0.0.Geometric Description of Spanv v22v v23v v2v v12v v1x x=

6、2v v1+3v v28 Finding a linear combination of given vectors 1,2,and 3to be equal to:The solution exists only when Span 1,2,3.Geometric Interpretationof Vector Equation6065551+5.55.06.02+1013=66747811+22+33=9 Recall:we defined matrix-matrix multiplications as the inner product between the row on the l

7、eft and the column on the right:e.g.,1634521121=13511193 Inspired by the vector equation,we can view as a linear combination of columns of the left matrix:605.51655.00556.01123=123123=11+22+33Matrix Multiplications as Linear Combinations of Vectors10 Linear combinations of columns Left matrix:bases,

8、right matrix:coefficients110101111123=1111+1012+0113One column on the rightMulti-columns on the right110101111112031=112233=123=1111+1012+0113=123=1111+1010+0111Matrix Multiplicationsas Column Combinations11 Linear combinations of rows of the right matrix Right matrix:bases,left matrix:coefficientsO

9、ne row on the left123110101111=1 110+2 101+3 111Matrix Multiplications as Row CombinationsMultiple rows on the left123101110101111=123123=TTT=123=1 110+2 101+3 111T=123=1 110+0 101+1 11112(Rank-1)outer product Sum of(Rank-1)outer productsMatrix Multiplicationsas Sum of(Rank-1)Outer Products111123=123123123111111123456=111123+111456=123123123+456456456Linear combination and spanExistence of solution of a linear systemMatrix multiplication as a linear combination of vectorsSummary