Answer to Question #205021 in Linear Algebra for Rohan Kumar

Question

let V={(a,b,c)€R³|a+b=c} and W={(a,b,c)€R³|a=b} be subspaces of R³.Is R³ direct sum of V and W?

Answer 1

Let $V=\{(a,b,c)\in\R^3\ |\ a+b=c\}$ and $W=\{(a,b,c)\in\R^3\ |\ a=b\}$ be subspaces of $\R^3$.

Since $V=\{(a,b,c)\in\R^3\ |\ a+b=c\}=\{(a,b,a+b)\in\R^3\ |\ a,b\in\R\}=\{a(1,0,1)+b(0,1,1)\in\R^3\ |\ a,b\in\R\},$

we conclude that the vector subspace $V$ is of dimension 2 with basis consisting of $(1,0,1)$ and $(0,1,1).$

Taking into account that $W=\{(a,b,c)\in\R^3\ |\ a=b\}=\{(a,a,c)\in\R^3\ |\ a,c\in\R\}=\{a(1,1,0)+c(0,0,1)\in\R^3\ |\ a,c\in\R\},$

we conclude that the vector subspace $W$ is of dimension 2 with basis consisting of $(1,1,0)$ and $(0,0,1).$

Since $\dim V+\dim W=2+2=4>3=\dim \R^3,$ we conclude that $\R^3$ can not be a direct sum of $V$ and $W.$