Quote:

Find the dimension and a basis of the following vector spaces V over the given field K.

$\displaystyle dim_{\mathbb{R}} \mathbb{R}^3=3$.Quote:

i). V is the set of all vectors $\displaystyle (\alpha,\beta,\gamma)$ in $\displaystyle \mathbb{R}^3$ with $\displaystyle \alpha+4\beta-3\gamma=0; \ K= \mathbb{R}$

Is the basis just $\displaystyle V=A(2,0,0)+B(0,1,0)+C(0,0,2)$?

I can't really see how to include the condition in a way better than this.

Once again $\displaystyle dim_{\mathbb{R}} \mathbb{R}^3=3$.Quote:

ii). V is the set of all vectors $\displaystyle (\alpha,\beta, \gamma)$ in $\displaystyle \mathbb{R}^3$ with $\displaystyle \alpha+\beta=\gamma$ and $\displaystyle \alpha-2\beta=-\gamma; \ k= \mathbb{R}$.

However, i'm a little puzzled regarding this basis. Would it be:

$\displaystyle V=\alpha(1,0,0) + \beta(0,2,0) + \gamma (0,0,3)$?