線性代數(考古題)

Question

Let Li(x1,x2,...,xn)=Σ (j=1,...,n)a(ij) x(j),a(ij)€R for i=1,2,...,m. Let
W={(b1,b2,...,bn)€R^n;Li(b1,b2,...,bn)=0,i=1,2,...,m}
be the subspace of R^n determined by the common zeros of the linear functionals L1,L2,...,Lm. Let f(x1,x2,...,xn) be a linear functional such that
f(b1,b2,...,bn)=0 for all (b1,b2,...,bn)€W. Prove or disprove that there exists
λ1, λ2,...,λm€R such that f=Σ(i=1,...,m)λiLi

Answer 1

令A為一m*n矩陣,A(i)表示矩陣A的第i列,A=[a]ij,for i=1,2,...,m,j=1,2..,n則L(i)(x1,x2,...,xn)=Σj=1na(ij)xj=,for i=1,2,...,mand L(i)(b)==0,for i=1,2,...,m,for all b€Wwe get:W⊥＝span{A1,A2,....,An}f(x) be a linear functional such that f(b)=0 for all b€WLet f(x1,x2,...,xn)=c1x1+c2x2+...+cnxn,c(i) are scalars for all i=>f(b)=c1b1+c2b2+...+cnbn==0,for all b€W=>c€W⊥=span{A1,A2,...,Am}so there exists λ1,λ2,...,λm€R such that c=λ1A1+...+λmAm=>f(x)==<λ1A1+...+λmAm,x>=λ1+...+λm=λ1L(1)(x)+..+λmL(m)(x)=Σi=1mλiL(i)(x)∴f=Σi=1mL(i)