# Homework1

**Topics:**Jacobian matrix and determinant, Partial derivative, Derivative

**Pages:**1 (268 words)

**Published:**October 4, 2014

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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