QBUS1040 Tutorial 2 Semester 2, 2022
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QBUS1040
Tutorial 2
Semester 2, 2022
Exercise 1: Transforming between two encodings for Boolean vectors
A Boolean n-vector is one for which all entries are either 0 or 1. Such vectors are used to encode whether each of n conditions holds, with ai = 1 meaning that condition i holds. Another common encoding of the same information uses the two values −1 and +1 for the entries. For example the Boolean vector (0, 1, 1, 0) would be written using this alternative encoding as ( −1, +1, +1, − 1). Suppose that x is a Boolean vector with entries that are 0 or 1, and y is a vector encoding the same information using the values −1 and +1. Express y in terms of x using vector notation. Also, express x in terms of y using vector notation.
Exercise 2: Word count and word count histogram vectors
Suppose the n-vector w is the word count vector associated with a document and dictionary of n words. For simplicity, we will assume that all words in the document appear in the dictionary.
(a) What is 1T w?
(b) What does w282 = 0 mean?
(c) Let h be the n-vector that gives the histogram of the word counts, i.e., hi is the fraction of the words in the document that are word i. Use vector notation to express h in terms of w . (You can assume that the document contains at least one word.)
Exercise 3: General formula for affine functions
Verify that formula (2.4) in VMLS holds for any affine function f : Rn → R. You can use the fact that f(x) = aT x + b for some n-vector a and scalar b.
Exercise 4: Python coding exercise
Using Python,
(a) Without using the numpy package, find xT y for the following data:
(i) x = (−1.2, 0.0, 3.7, −7.2), y = (4, 3, 2, 1)
(ii) x = (1, 2, 3, . . . , 9), y = (9, 8, 7, . . . , 1)
(b) find the maximum and minimum entry in a random vector of length 20 and compare your result to the inbuilt max() and min() functions.
Exercise 5:
How many flops does it take to form a linear combination of 100 vectors of length 105 (with 100 nonzero coefficients)? About how long would this take on a computer capable of carrying out 1 Gflop/s?
2022-11-04