# LING570 Hw9: Math basics solution

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Q1 (8 points): Logistic function: (see https://en.wikipedia.org/wiki/Logistic function)
1a (3 pts): What’s a logistic function? Write down the formula.
1b (2 pts): What is the standard logistic function? Write down the formula.
1c (3 pts): What is the derivative of the standard logistic function f(x)? Write down the formula.

Q2 (8 points): Sigmoid function: See https://en.wikipedia.org/wiki/Sigmoid function

2a (4 pts): What is a sigmoid function? Write down the definition.
2b (4 pts): What is the relation between sigmoid function and logistic function?

Q3 (8 points): Tanh function: (see https://en.wikipedia.org/wiki/Hyperbolic function and
https://en.wikipedia.org/wiki/Sigmoid function)
3a (4 pts): Write down the formula for the tanh function.
3b (4 pts): What is the relation between sigmoid function and tanh function?

Q4 (8 points): The softmax function:
4a (4 pts): What is the softmax function? Write down the formula.
See https://en.wikipedia.org/wiki/Softmax function

4b (4 pts): If a vector x is [1, 2, 3], what is the value of softmax(x)?

Q5 (18 points): Matrix: see Sect 1-3 of https://en.wikipedia.org/wiki/Matrix (mathematics)
5a (12 points): Let A =

1 2 3
4 5 6
7 8 9

 and B =

2
1
3

what is A × B?

what is B × A?
what is the transpose of A?
what is the tranpose of B?
what are the dimensions of B?
what are the dimensions of the transpose of B?

5b (6 points): Let A =

1 2
3 4
and B =

2 0
1 2
.
What is A × B?
what is B × A?
Is matrix multiplication communtative?

Submission: submit only one file named hw.pdf or hw.jpeg to Canvas.