Compute $\mathbf{u}+\mathbf{v}$ and $\mathbf{u}-3 \mathbf{v}$
\[
\mathbf{u}=\left[\begin{array}{r}
-1 \\
2
\end{array}\right], \mathbf{v}=\left[\begin{array}{r}
-2 \\
1
\end{array}\right]
\]

Step 1 ：Given vectors are \(\mathbf{u} = \left[\begin{array}{r} -1 \\ 2 \end{array}\right]\) and \(\mathbf{v} = \left[\begin{array}{r} -2 \\ 1 \end{array}\right]\)

Step 2 ：To compute \(\mathbf{u} + \mathbf{v}\), we add the corresponding components of the two vectors, which gives us \(\mathbf{u} + \mathbf{v} = \left[\begin{array}{r} -3 \\ 3 \end{array}\right]\)

Step 3 ：To compute \(\mathbf{u} - 3\mathbf{v}\), we first multiply the vector \(\mathbf{v}\) by the scalar -3, and then subtract the resulting vector from \(\mathbf{u}\), which gives us \(\mathbf{u} - 3\mathbf{v} = \left[\begin{array}{r} 5 \\ -1 \end{array}\right]\)

Step 4 ：So, the final answers are \(\boxed{\left[\begin{array}{r} -3 \\ 3 \end{array}\right]}\) and \(\boxed{\left[\begin{array}{r} 5 \\ -1 \end{array}\right]}\)