1. ## subspace 1

Check if S is subspace of V

1) S is the set of all upper triangular matrices. V = Mnxm

2) S is the set of all the matrices nxn whose the determinant is 1. V = Mnxn

2. Check my solution

1)
$\displaystyle u = \begin{pmatrix} a11 & a21 & a31 \\ 0 & b21 & c31 \\ 0 & 0 & c31\end{pmatrix}$

$\displaystyle v= \begin{pmatrix} a12 & a22 & a32 \\ 0 & b22 & c32 \\ 0 & 0 & c32\end{pmatrix}$

$\displaystyle u+v=\begin{pmatrix} (a11+a12) & (a21+a22) & (a31+a32) \\ 0 & (b11+b22) & (b31+b32) \\ 0 & 0 & (c31+c32)\end{pmatrix}$

u+v belongs to S

$\displaystyle \alpha . u = \begin{pmatrix} \alpha a11 & \alpha a21 & \alpha a31 \\ 0 & \alpha b21 & \alpha b31 \\ 0 & 0 & \alpha c31 \end{pmatrix}$

$\displaystyle \alpha . u$ belongs to S

2)
$\displaystyle \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$(determinant equal to 1)

$\displaystyle u = \begin{pmatrix} a1 & 0 \\ 0 & b1 \end{pmatrix}$

$\displaystyle v = \begin{pmatrix} a2 & 0 \\ 0 & b2 \end{pmatrix}$

$\displaystyle u+v=\begin{pmatrix} a1+a2 & 0 \\ 0 & b1+b2 \end{pmatrix}$

u+v It belong to S

$\displaystyle \alpha . u = \begin{pmatrix} \alpha a1 & 0 \\ 0 & \alpha b2\end{pmatrix}$

$\displaystyle \alpha . u$ belong to S