Homework # 3
due February 15, 3:30 PM

Vectors

Mathematical vectors in $n$-dimensional Euclidean space can be represented by lists of real numbers. For this homework, you will define several operations on vectors:

1. Write a function scale that takes a real and a list of reals and returns a list which multiples the number to every element in the list. For example:

   - scale (3.0,[1.0,2.0,3.0]) ;
   val it = [3.0,6.0,9.0] : real list
   

2. Write a function dot that takes two lists of real numbers of the same length and returns the ``dot product'' (the sum of the element-wise multiplication, also known as the ``inner product''). For example:

   - dot ([1.0, 2.0],[~3.0, 4.0, 6.0]) ;
   val it = 5.0 : real
   

   As seen here, your program should assume a shorter vector is extended with zeros as necessary.
3. Write a function magn that takes a list of real numbers and returns the square root of the sum of the squares of the elements. Hint, you can use dot to do most of the work.

   - magn [1.0,1.0,1.0,2.0,3.0] ;
   val it = 4.0 : real
   

   (You will need to ``open Math'' to make the sqrt function available.)
4. Define an addition function for vectors:

   - add ([1.0, 2.0], [~5.0, 0.0, 1.5]) ;
   val it = [~4.0, 2.0, 1.5] : real list
   

Put all your functions in a single file named vector.sml in your AFS volume under directory homework3.

Types

Do Exercises 1 and 2 of Chapter 6 (pages 101-102). Write your answers on paper.

Submitting Your Work

Turn in the first part in AFS, and the second part on paper at the beginning of lecture.

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