Homework1

Topics: Jacobian matrix and determinant, Partial derivative, Derivative Pages: 1 (268 words) Published: October 4, 2014
ISyE 3133B Spring 2011

Assignment 1
Assigned on Friday 1/14/2011
Due by 10:00am on Wednesday 1/19/2011

1. Expand the following summations:
(For example, the answer to part (a) is x1 + x2 + x3 .)
(a)
(c)

3
i=1 xi
3
t
t=1 2 w2t

(b)
(d)

3
i=1
5
n=3

4
j=2 (xi + yij )
n+3
m=n+1 xn ym






2
3
2. Consider the following two vectors: x =  1  and y =  1 . 4
2
(a) What is dimension of x and y?
(b) Compute x + 2y.
(c) Compute ||x − y||.
(d) Compute xT (x + y).
(e) Compute x2 y1 .
3. Consider the following system of equations with unknowns x1 , x2 and x3 : x1 = 3x2 − 2x1 + 4 − x3
x2 + 2x1 = 4
4x1 + x2 − 3x3 = 6 − x2
Express this system in matrix vector notation in the form of Ax = b
where x = (x1 , x2 , x3 ) is the vector of unknowns. What is the matrix A and vector b? Compute A−1 (you can use a calculator) if it exists and solve the system if it is solvable. 4. Let A be an invertible m × m matrix, b be an m × 1 vector, C be an m × n matrix, and d be an m × 1 vector. What is the dimension of the following expression:

(b + A−1 d) C ?
5. Consider the function
f (x, y) =

x2
.
1 + xy

Write an expression for its gradient f (x, y), i.e. the vector of partial derivatives. What is the dimension of the gradient vector? Evaluate the gradient at the point (1, 2).

1

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