Scalar multiplication

Find $-3\left[\begin{array}{r}5\\ -1\\ 2\\ -3\end{array}\right]+5\left[\begin{array}{r}-8\\ 2\\ -3\\ 6\end{array}\right]$.

Use the properties of scalar multiplication from Proposition 2.17 and the properties of vector addition from Proposition 2.12 to prove the following equalities. Justify every step.

1. $(k+\ell)(\mathbf{u}+\mathbf{v})=k\mathbf{u}+k\mathbf{v}+\ell\mathbf{u}+\ell\mathbf{v}$.

2. $0\mathbf{u}=\mathbf{0}$.

3. $(-1)\mathbf{u}=-\mathbf{u}$.

4. $-(k\mathbf{u})=k(-\mathbf{u})=(-k)\mathbf{u}$.