判断以下命题的错对,去证明你的答案!a)Thesetofallvectorsoftheform(a,b,c)withb=a+cisasubspaceofR3.b)Thesetofalln*nmatricesAsuchthatdet(A)=0isasubspaceofMnn.c)Thesetofallpolynomialsoftheforma0+a1x+a2x^2

Question

判断以下命题的错对,去证明你的答案!
a)The set of all vectors of the form(a,b,c) with b=a+c is a subspace of R3.
b)The set of all n*n matrices A such that det(A)=0 is a subspace of Mnn.
c)The set of all polynomials of the form a0+a1x+a2x^2+a3x^3 in which a0,a1,a2,a3,are integers is a subspace of P3.
大概翻译如下：
a）所有的向量集形式（a,b,c）与b= a+ c是R3的子空间.
b）所有的n* n矩阵的集合,这样一个det（A）= 0是一个子空间Mnn.
c）A0形式的所有多项式集+ a1x a2x^2+ a3x^3,其中a0,a1,a2,a3,都是整数是P3的子空间.

Answer 1

a)right
proof:
Let D denote the set of all vectors of the form (a,b,c) with b=a+c.
Take two arbitrary elements from D ,denoted by p1=(a1,b1,c1) and p2=(a2,b2,c2) .
For any real numbers x and y,we have xb1+yb2=x(a1+c1)+y(a2+c2)=(xa1+xc1)+(ya2+yc2)
Then xp1+yp2=x(a1,b1,c1)+y(a2,b2,c2)=(xa1+ya2,xb1+yb2,xc1+yc2) is in D.
Hence D is a subspace of R3.
b)wrong
proof:
Give a counterexample:
let A=[1,1,...,1;0,0,...,0;.;0,0,...,0] and B=[0,0,...,0;1,2,...,n;2,3,...,n,1;3,4,...,n,1,2;...;n-1,n,1,...,n-2]
Thus det(A)=det(B)=0,but det(A+B)≠0.
Hence the set of all n*n matrices A such that det(A)=0 is not a subspace of Mnn.
c)wrong
Denote the set of all polynomials of the form a0+a1x+a2x^2+a3x^3 in which a0,a1,a2,a3,are integers by D.
let f=a0+a1x+a2x^2+a3x^3 and g=b0+b1x+b2x^2+b3x^3
take p=1/(2a0) and q=1/(3b0)
then the constant term of pf+qg=1/6
so pf+qg is not in D,whence D is not a subspace of P3.