Answer to Question #235670 in Linear Algebra for Amuj

Question #235670

Let the vector space V=R^3 and W={(a,b,c);a+b+c=0} i.e. W consists of those vectors each with the property that the sum of its components is zero. Is W a subspace of V


1
Expert's answer
2021-09-13T07:43:26-0400

Let us use the subspace criterion: a nonempty subset "W\\subset V" is a vector subspace of the space "V" over the field "R" , if and only if the following statements are true: "1) \\forall u, v\\in W (u+v)\\in W;" "2) \\forall\\alpha\\in R \\forall u \\in W (\\alpha u)\\in W" .

Let's check these statements for the case "V=R^3"  and "W=\\{(a,b,c);a+b+c=0\\}"

1) "\\forall u(a;b;c), v(m;n;k)\\in W" "(u+v)=(a+m;b+n;c+k)" and "a+m+b+n+c+k=a+b+c+m+n+k=0+0=0" . So "(u+v)\\in W"

2) "\\forall\\alpha\\in R \\forall u (a;b;c) \\in W" "(\\alpha u)= (\\alpha a; \\alpha b; \\alpha c)" and "\\alpha a+\\alpha b+\\alpha c =\\alpha (a+b+c)=0" .

So "(\\alpha u)\\in W" . So "W" is a vector subspace of the space "V"

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