CS-584 - Homework 0 (0%)

• Let: A = ⎡⎢⎢⎢⎣ 1   2   3 ⎤⎥⎥⎥⎦, B = ⎡⎢⎢⎢⎣ 4   5   6 ⎤⎥⎥⎥⎦, C = ⎡⎢⎢⎢⎣  − 1   1   3 ⎤⎥⎥⎥⎦, find:
  1. 2A − B
  2. ||A|| and the angle of A relative to the positive X axis
  3. Â, a unit vector in the direction of A
  4. the direction cosines of A
  5. A⋅B and B⋅A
  6. the angle between A and B
  7. a vector which is perpendicular to A
  8. A × B and B × A
  9. a vector which is perpendicular to both A and B
  10. the linear dependency between A, B, C
  11. A^TB and AB^T.
• 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:
  1. 2A − B
  2. AB and BA
  3. (AB)^T and B^TA^T
  4. |A| and |C| (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)
• Let: A = ⎡⎢⎣ 1 2     3 2 ⎤⎥⎦, B = ⎡⎢⎣ 2  − 2      − 2 5 ⎤⎥⎦, find:
  1. the eigenvalues and corresponding eigenvectors of A.
  2. the matrix V^− 1AV 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).
• Let: f(x) = x²  + 3,   g(x,  y) = x² + y², find:
  1. the first and second derivatives of f(x) with respect to x: f’(x), and f’’(x).
  2. the partial derivatives: (∂g)/(∂x), and (∂g)/(∂y).
  3. the gradient vector ∇g(x,  y).
  4. the probability density function (pdf) of a univariate Gaussian (normal) distribution.

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