Sample Vector and Matrix Arithmetic Questions
-
Write out the 4x4 identity matrix.
-
Write out the 4x4 null matrix.
-
Evaluate each of the following:
- \(\left[ \begin{array}{r}5 \\ 2 \\ 8 \end{array}\right] +
\left[ \begin{array}{r}2 \\ -1 \\ 3 \end{array}\right]\)
- \(\left[ \begin{array}{r}5 \\ 2 \\ 8 \end{array}\right] \cdot
\left[ \begin{array}{r}2 \\ 3 \\ -2 \end{array}\right]\)
- \(\left\Vert [3 \; 4 \; 8] \right\Vert\)
- \(\left\Vert \frac{[3 \; 4 \; 8]}{|| [3 \; 4 \; 8] ||} \right\Vert\)
- \(
\left[ \begin{array}{r r r}
1 & 2 & 3 \\
4 & 5 & 6 \\
\end{array}\right]^{T}
\)
- \(
\left[ \begin{array}{r r r}
1 & 2 & 3 \\
4 & 5 & 6 \\
\end{array}\right] +
\left[ \begin{array}{r r r}
11 & 12 & 13 \\
14 & 15 & 16 \\
\end{array}\right]
\)
- \(
\left[ \begin{array}{r r r}
1 & 2 & 3 \\
4 & 5 & 6 \\
\end{array}\right]
\left[ \begin{array}{r}
2 \\
3 \\
1
\end{array}\right]
\)
- \(
\left[ \begin{array}{r r r}
1 & 2 & 3 \\
4 & 5 & 6 \\
\end{array}\right]
\left[ \begin{array}{r r}
3 & 1 \\
7 & 5\\
2 & 4
\end{array}\right]
\)
- \(
\left[ \begin{array}{r r r}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9 \\
\end{array}\right]
\left[ \begin{array}{r r r}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{array}\right]
\)
- \(
\left|
\left[ \begin{array}{r r}
5 & 3 \\
7 & 9 \\
\end{array}\right]
\right|
\)
-
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 \(\bs{B} = \bs{C}\)?
Copyright
2014
