Interactive components, just for testing, no commentary.
Exercises15.1True/False
A True/False question.
1.True/False.
Every vector space has finite dimension.
True.
The vector space of all polynomials with finite degree has a basis, \(B = \{1,x,x^2,x^3,\dots\}\text{,}\) which is infinte.
False.
The vector space of all polynomials with finite degree has a basis, \(B = \{1,x,x^2,x^3,\dots\}\text{,}\) which is infinte.
Hint.
\(P_n\text{,}\) the vector space of polynomials with degree at most \(n\text{,}\) has dimension \(n+1\) by 2.1. [Cross-reference is just a demo, content is not relevant.] What happens if we relax the defintion and remove the parameter \(n\text{?}\)
Exercises15.2Multiple-Choice
Multiple-Choice problem
1.Multiple-Choice, Not Randomized, One Answer.
What color is a stop sign?
Green
Green means “go!”.
Red
Red is universally used for prohibited activities or serious warnings.
White
White might be hard to see.
Hint1.
What did you see last time you went driving?
Hint2.
Maybe go out for a drive?
Exercises15.3Parsons Problem, Math Proof
With some MathJax.
1.Parsons Problem, Mathematical Proof.
Create a proof of the theorem: If \(n\) is an even number, then \(n\equiv 0\mod 2\text{.}\)
Suppose \(n\) is even.
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Then there exists an \(m\) so that \(n = 2m\text{.}\)
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Then \(n\) is a prime number.
#paired
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Then there exists an \(m\) so that \(n = 2m + 1\text{.}\)
#paired
---
Click the heels of your ruby slippers together three times.
#distractor
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So we have the displayed equation:
\begin{equation*}
n = 2m + 0\text{.}
\end{equation*}
This is a superfluous second paragraph in this block.
The Sieve of Eratosthenes computes prime numbers by starting with a finite list of the integers bigger than 1. The first member of the list is a prime and is saved/recorded. Then all multiples of that prime (which not a prime, excepting the prime itself!) are removed from the list. Now the first number remaining in the list is the next prime number. And the process repeats.
The code blocks below can be rearranged to form one of the many possible programs to implement this algorithm to compute a list of all the primes less than \(250\text{.}\) [Ed. this version of this problem requires the reader to provide the necessary indentation.]
n = 250
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primes = []
candidates = list(range(2,n))
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candidates = []
primes = list(range(2,n)) #paired
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primes = candidates + [p] #distractor
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while candidates:
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p = candidates[0]
primes.append(p)
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for nonprime in range(p, n, p):
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if nonprime in candidates:
candidates.remove(nonprime)
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print(primes)
Exercises15.5Matching
Events and dates.
1.Matching Problem, Dates.
Match each event in United States history with the year it happened.
