Problem
Determine whether the indicated subset is a subspace of the given euclidean space:
$ \{[x,y,z]\ |\ x,y,z \in \mathbb{R} $ and $z=3x+2\}$ in $\mathbb{R}^{3}$
Solution
By definition, in order for a subset to be a subspace 3 conditions must be occur:
To pass by the originTo contain the origin.- To be closed under addition.
- To be closed under scalar multiplication.
So I try to solve the exercise by this way:
$1.$ The origin $(0,0,0) \in \mathbb{W} $
$2.$ Let $\vec u$ and $\vec v \in \mathbb{W} $. We have
$ \begin{cases} 3u_1 + 2 - u_3 = 0 \\ 3v_1 + 2 - v_3 = 0 \\ \end{cases} $ The sum is $ 6(u_1 + v_1) + 4 - (u_3+v_3) = 0 $ (which $\in \mathbb{W} $)
$3.$ Let $\vec u$ $ \in \mathbb{W} $ and $\ r$ $ \in \mathbb{R} $. We have
$r(3u_1) + r(2) - r(u_3) = 0 \\$
Which, also, $ \in \mathbb{W} $
So, why is the book's answer: It isn't a subspace?