Sample Vector and Matrix Arithmetic Questions

Write out the 4x4 identity matrix.
Write out the 4x4 null matrix.
Evaluate each of the following:
1. \(\left[ \begin{array}{r}5 \\ 2 \\ 8 \end{array}\right] + \left[ \begin{array}{r}2 \\ -1 \\ 3 \end{array}\right]\)
2. \(\left[ \begin{array}{r}5 \\ 2 \\ 8 \end{array}\right] \cdot \left[ \begin{array}{r}2 \\ 3 \\ -2 \end{array}\right]\)
3. \(\left\Vert [3 \; 4 \; 8] \right\Vert\)
4. \(\left\Vert \frac{[3 \; 4 \; 8]}{|| [3 \; 4 \; 8] ||} \right\Vert\)
5. \( \left[ \begin{array}{r r r} 1 & 2 & 3 \\ 4 & 5 & 6 \\ \end{array}\right]^{T} \)
6. \( \left[ \begin{array}{r r r} 1 & 2 & 3 \\ 4 & 5 & 6 \\ \end{array}\right] + \left[ \begin{array}{r r r} 11 & 12 & 13 \\ 14 & 15 & 16 \\ \end{array}\right] \)
7. \( \left[ \begin{array}{r r r} 1 & 2 & 3 \\ 4 & 5 & 6 \\ \end{array}\right] \left[ \begin{array}{r} 2 \\ 3 \\ 1 \end{array}\right] \)
8. \( \left[ \begin{array}{r r r} 1 & 2 & 3 \\ 4 & 5 & 6 \\ \end{array}\right] \left[ \begin{array}{r r} 3 & 1 \\ 7 & 5\\ 2 & 4 \end{array}\right] \)
9. \( \left[ \begin{array}{r r r} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ \end{array}\right] \left[ \begin{array}{r r r} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array}\right] \)
10. \( \left| \left[ \begin{array}{r r} 5 & 3 \\ 7 & 9 \\ \end{array}\right] \right| \)
Illustrate both ways of visualizing vector subtraction. That is, draw two points \(\bs{p}\) and \(\bs{q}\) and then draw the resulting point \(\bs{p}-\bs{q}\) and the translated direction vector \(\bs{p}-\bs{q}\).
Explain the process of normalizing a vector. What is the norm of a normalized vector?
How can you determine if two vectors are orthogonal? Are the vectors \([2 \; 2]\) and \([-5 \; 5]\) orthogonal?
Illustrate the set of convex combinations of two points, \(\bs{p}\) and \(\bs{q}\).
Given three matrices such that \(\bs{A} \bs{B} = \bs{A} \bs{C}\). Does it follow that \(\bs{B} = \bs{C}\)?