Vector Spaces and Subspaces

  1. Check if the set \(H= \left\{\begin{bmatrix}3s\\2s\\3s-2\end{bmatrix}: s \in \mathbb{R}\right\}\) is a subspace of \(\mathbb{R}^3 \). Justify your answer.


  2. Is the following set a subspace of \(\mathbb{P}_2\). All polynomials of the form \(p(t)=a+bt^2\), where \(a, b\in \mathbb{R}\).


  3. Determine if the set \(H\) of all matrices of the form \(\left[\begin{array}{ll}a & b \\ 0 & -a\end{array}\right]\) is a subspace of \(M_{2 \times 2}\), the vector space of all \(2\times 2\) matrices.

    4.  

Soution of problems 1-3.