**Due by: September 1, 2015
**

- Let:
*A*= ⎡⎢⎢⎢⎣ 1 2 3 ⎤⎥⎥⎥⎦,*B*= ⎡⎢⎢⎢⎣ 4 5 6 ⎤⎥⎥⎥⎦,*C*= ⎡⎢⎢⎢⎣ − 1 1 3 ⎤⎥⎥⎥⎦, find:- 2
*A*−*B* - ||
*A*|| and the angle of*A*relative to the positive*X*axis -
*Â*, a unit vector in the direction of*A* - the direction cosines of
*A* -
*A*⋅*B*and*B*⋅*A* - the angle between
*A*and*B* - a vector which is perpendicular to
*A* -
*A*×*B*and*B*×*A* - a vector which is perpendicular to both
*A*and*B* - the linear dependency between
*A*,*B*,*C* -
*A*^{T}*B*and*A**B*^{T}.

- 2
- Let:
*A*= ⎡⎢⎢⎢⎣ 1 2 3 4 − 2 3 0 5 − 1 ⎤⎥⎥⎥⎦,*B*= ⎡⎢⎢⎢⎣ 1 2 1 2 1 − 4 3 − 2 1 ⎤⎥⎥⎥⎦,*C*= ⎡⎢⎢⎢⎣ 1 2 3 4 5 6 − 1 1 3 ⎤⎥⎥⎥⎦, find:- 2
*A*−*B* -
*AB*and*BA* - (
*AB*)^{T}and*B*^{T}*A*^{T} - |
*A*| and |*C*| (note A-10) - the matrix (
*A*,*B*, or*C*) in which the row vectors form an orthogonal set -
*A*^{ − 1}and*B*^{ − 1}(note B-5)

- 2
- Let:
*A*= ⎡⎢⎣ 1 2 3 2 ⎤⎥⎦,*B*= ⎡⎢⎣ 2 − 2 − 2 5 ⎤⎥⎦, find:- the eigenvalues and corresponding eigenvectors of
*A*. - the matrix
*V*^{ − 1}*AV*where*V*is composed of the eigenvectors of*A*. - the dot product between the eigenvectors of
*A*. - the dot product between the eigenvectors of
*B*. - the property of the eigenvectors of
*B*and its reason (note C-4).

- the eigenvalues and corresponding eigenvectors of
- Let:
*f*(*x*) =*x*^{2}+ 3,*g*(*x*,*y*) =*x*^{2}+*y*^{2}, find:- the first and second derivatives of
*f*(*x*) with respect to*x*:*f*’(*x*), and*f*’’(*x*). - the partial derivatives: (∂
*g*)/(∂*x*), and (∂*g*)/(∂*y*). - the gradient vector ∇
*g*(*x*,*y*). - the probability density function (pdf) of a univariate Gaussian (normal) distribution.

- the first and second derivatives of