CS-511 - Homework 0 (0%)

Due by: August 28, 2008

A.

Let: $A = \left[\begin{array}{r} 1\\ 2\\ 3\end{array}\right]$, $B = \left[\begin{array}{r} 4\\ 5\\ 6\end{array}\right]$, $C = \left[\begin{array}{r} -1\\ 1\\ 3\end{array}\right]$, find:

1. $2A - B$
2. $\vert\vert A\vert\vert$ and the angle of $A$ relative to the positive $X$ axis
3. $\hat{A}$, a unit vector in the direction of $A$
4. the direction cosines of $A$
5. $A \cdot B$ and $B \cdot A$
6. the angle between $A$ and $B$
7. a vector which is perpendicular to $A$
8. $A \times B$ and $B \times A$
9. a vector which is perpendicular to both $A$ and $B$
10. the linear dependency between $A$, $B$, $C$
11. $A^T B$ and $A B^T$.

B.

Let: $A = \left[\begin{array}{rrr} 1 & 2 & 3\\ 4 & -2 & 3\\ 0 & 5 & -1 \end{array}\right]$, $B = \left[\begin{array}{rrr} 1 & 2 & 1\\ 2 & 1 & -4\\ 3 & -2 & 1 \end{array}\right]$, $C = \left[\begin{array}{rrr} 1 & 2 & 3\\ 4 & 5 & 6\\ -1 & 1 & 3 \end{array}\right]$, find:

1. $2A - B$
2. $AB$ and $BA$
3. $(AB)^{T}$ and $B^{T}A^{T}$
4. $\vert A\vert$ and $\vert C\vert$ (note A-10)
5. the matrix ($A$, $B$, or $C$) in which the row vectors form an orthogonal set
6. $A^{-1}$ and $B^{-1}$ (note B-5)

C.

Let: $A = \left[\begin{array}{rr} 1 & 2 \\ 3 & 2 \end{array}\right]$, $B = \left[\begin{array}{rr} 2 & -2 \\ -2 & 5 \end{array}\right]$, find:

1. the eigenvalues and corresponding eigenvectors of $A$.
2. the matrix $V^{-1}AV$ where $V$ is composed of the eigenvectors of $A$.
3. the dot product between the eigenvectors of $A$.
4. the dot product between the eigenvectors of $B$.
5. the property of the eigenvectors of $B$ and its reason (note C-4).

D.

Let: $f(x)=x^2+3$,   $g(x,y)=x^2+y^2$, find:

1. the first and second derivatives of $f(x)$ with respect to $x$: $f'(x)$, and $f''(x)$.
2. the partial derivatives: $\frac{\partial g}{\partial x}$, and $\frac{\partial g}{\partial y}$.
3. the gradient vector $\nabla g(x,y)$.
4. the probability density function (pdf) of a univariate Gaussian (normal) distribution.