1
10 points
Which of the following can be used to conclude that \( W=\left\{(x, y, z)^{T} \in \mathbb{R}^{3} \mid x-2 y+z^{2}=0\right\} \) does not form a subspace of \( \mathbb{R}^{3} \) ?
\( (0,0,0)^{T} \notin W \)
\( (1,1,1)^{T}+(-1,1,2)^{T} \notin W \)
\( (2,1,0)^{T}+(4,2,0)^{T} \notin W \)
\( (1,1,1)^{T}+(0,2,2)^{T} \notin W \)
2
10 points
Which of the following constraints results in the set of all polynomials of the form \( a_{0}+a_{1} x+a_{2} x^{2} \) forms a subspace of \( \mathcal{P}_{2}(\mathbb{R}) \) ?
Select all the correct options.
\( a_{0}=a_{1}=a_{2}=0 \)
\( a_{1}=0 \)
\( a_{0} \geq 0 \)
\( 2 a_{1}=a_{2} \)
