Question

Example of Vector space and subspace in which it's all properties must satisfied.

Answer 1

Consider the vector space $\R^3$ over field $\R$ with usual addition

$(x_1,y_1,z_1)+(x_2,y_2,z_2)=(x_1+x_2,y_1+y_2,z_1+z_2)$

and multiplication by a scalar $\alpha\in\R:\ \alpha\cdot(x,y,z)=(\alpha x, \alpha y, \alpha z).$

Let us show that $W= \{(x,-3x,2x)|x\in\mathbb R\}$ is a subspace of $\mathbb R^3$. Let $\alpha, \beta\in\mathbb R,\ (x,-3x,2x), (y,-3y,2y)\in W.$ Then $\alpha (x,-3x,2x)+\beta (y,-3y,2y)= (\alpha x,-3\alpha x,2\alpha x)+(\beta y,-3\beta y,2\beta y)= (\alpha x+\beta y,-3\alpha x-3\beta y,2\alpha x+2\beta y)= (\alpha x+\beta y,-3(\alpha x+\beta y),2(\alpha x+\beta y))\in W.$

Therefore, $W= \{(x,-3x,2x)|x\in\mathbb R\}$ is a subspace of $\mathbb R^3$.