Scale for project Matrix

Introduction

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Guidelines

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- Check carefully that no malicious aliases was used to fool you
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official repository.

- To avoid any surprises, carefully check that both the evaluating
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- Use the flags available on this scale to signal an empty repository,
non-functioning program, a norm error, cheating etc. In these cases,
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cheating). However, with the exception of cheating, you are
encouraged to continue to discuss your work (even if you have not
finished it) in order to identify any issues that may have caused
this failure and avoid repeating the same mistake in the future.

- Remember that for the duration of the defence, no segfault,
no other unexpected, premature, uncontrolled or unexpected
termination of the program, else the final grade is 0. Use the
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If you want to edit a file, take the time to explicit the reasons with the
evaluated student and make sure both of you are okay with this.

- You must also verify the absence of memory leaks. Any memory allocated on the heap must
be properly freed before the end of execution.
You are allowed to use any of the different tools available on the computer, such as
leaks, valgrind, or e_fence. In case of memory leaks, tick the appropriate flag.

Exercise 00 - Add, Subtract and Multiply

Complexity

Ask the student to justify the complexity of the functions. It must be at
most O(n) in time and O(n) in space.

Check the use of forbidden mathematical functions (see the subject).

Add

Check the behaviour of the vector addition with the following parameters:

  • '[0, 0]' and '[0, 0]' give '[0, 0]'
  • '[1, 0]' and '[0, 1]' give '[1, 1]'
  • '[1, 1]' and '[1, 1]' give '[2, 2]'
  • '[21, 21]' and '[21, 21]' give '[42, 42]'
  • '[-21, 21]' and '[21, -21]' give '[0, 0]'
  • '[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]' and '[9, 8, 7, 6, 5, 4, 3, 2, 1, 0]' give '[9, 9, 9, 9, 9, 9, 9, 9, 9, 9]'

Check the behaviour of the matrix addition with the following parameters:

  • '[[0, 0], [0, 0]]' and '[[0, 0], [0, 0]]' give '[[0, 0], [0, 0]]'
  • '[[1, 0], [0, 1]]' and '[[0, 0], [0, 0]]' give '[[1, 0], [0, 1]]'
  • '[[1, 1], [1, 1]]' and '[[1, 1], [1, 1]]' give '[[2, 2], [2, 2]]'
  • '[[21, 21], [21, 21]]' and '[[21, 21], [21, 21]]' give '[[42, 42], [42, 42]]'

Feel free to perform more tests on your own

Subtract

Check the behaviour of vector subtraction with the following parameters:

  • '[0, 0]' and '[0, 0]' give '[0, 0]'
  • '[1, 0]' and '[0, 1]' give '[1, -1]'
  • '[1, 1]' and '[1, 1]' give '[0, 0]'
  • '[21, 21]' and '[21, 21]' give '[0, 0]'
  • '[-21, 21]' and '[21, -21]' give '[-42, 42]'
  • '[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]' and '[9, 8, 7, 6, 5, 4, 3, 2, 1, 0]' give '[-9, -7, -5, -3, -1, 1, 3, 5, 7, 9]'

Check the behaviour of matrix subtraction with the following parameters:

  • '[[0, 0], [0, 0]]' and '[[0, 0], [0, 0]]' give '[[0, 0], [0, 0]]'
  • '[[1, 0], [0, 1]]' and '[[0, 0], [0, 0]]' give '[[1, 0], [0, 1]]'
  • '[[1, 1], [1, 1]]' and '[[1, 1], [1, 1]]' give '[[0, 0], [0, 0]]'
  • '[[21, 21], [21, 21]]' and '[[21, 21], [21, 21]]' give '[[0, 0], [0, 0]]'

Feel free to perform more tests on your own

Multiply

Check the behaviour of vector scaling with the following parameters:

  • '[0, 0]' and '1' give '[0, 0]'
  • '[1, 0]' and '1' give '[1, 0]'
  • '[1, 1]' and '2' give '[2, 2]'
  • '[21, 21]' and '2' give '[42, 42]'
  • '[42, 42]' and '0.5' give '[21, 21]'

Check the behaviour of matrix scaling with the following parameters:

  • '[[0, 0], [0, 0]]' and '0' give '[[0, 0], [0, 0]]'
  • '[[1, 0], [0, 1]]' and '1' give '[[1, 0], [0, 1]]'
  • '[[1, 2], [3, 4]]' and '2' give '[[2, 4], [6, 8]]'
  • '[[21, 21], [21, 21]]' and '0.5' give '[[10.5, 10.5], [10.5, 10.5]]'

Feel free to perform more tests on your own

Exercise 01 - Linear combination

Complexity

Ask the student to justify the complexity of the function. It must be at
most O(n) in time and O(n) in space.

Check the use of forbidden mathematical functions (see the subject).

Basic tests

Test the behaviour of linear combinations of vectors with the following parameters:

  • 'linear_combination([Vector::from([-42., 42.])], [-1.])' gives '[42., -42.]'
  • 'linear_combination([Vector::from([-42.]), Vector::from([-42.]), Vector::from([-42.])], [-1., 1., 0.])' gives '[0.]'
  • 'linear_combination([Vector::from([-42., 42.]), Vector::from([1., 3.]), Vector::from([10., 20.])], [1., -10., -1.])' gives '[-62., -8.]'
  • 'linear_combination([Vector::from([-42., 100., -69.5]), Vector::from([1., 3., 5.])], [1., -10.])' gives '[-52., 70., -119.5]'

Feel free to perform more tests on your own.

Exercise 02 - Linear interpolation

Complexity

Ask the student to justify the complexity of the function. It must be at
most O(n) in time and O(n) in space.

Check the use of forbidden mathematical functions (see the subject).

Basic tests

Check the behaviour of the function with the following parameters:

  • 'lerp(0., 1., 0.)' gives '0.'
  • 'lerp(0., 1., 1.)' gives '1.'
  • 'lerp(0., 42., 0.5)' gives '21.'
  • 'lerp(-42., 42., 0.5)' gives '0.'
  • 'lerp(Vector::from([-42., 42.]), Vector::from([42., -42.]), 0.5)' gives '[0.0] [0.0]'

Feel free to perform more tests on your own.

Exercise 03 - Dot product

Complexity

Ask the student to justify the complexity of the function. It must be at
most O(n) in time and O(n) in space.

Check the use of forbidden mathematical functions (see the subject).

Basic tests

Check the behaviour of the function with the following parameters:

  • '[0, 0]' and '[0, 0]' gives '0'
  • '[1, 0]' and '[0, 0]' gives '0'
  • '[1, 0]' and '[1, 0]' gives '1'
  • '[1, 0]' and '[0, 1]' gives '0'
  • '[1, 1]' and '[1, 1]' gives '2'
  • '[4, 2]' and '[2, 1]' gives '10'

Feel free to perform more tests on your own.

Exercise 04 - Norm

Complexity

Ask the student to justify the complexity of the functions. It must be at
most O(n) in time and O(n) in space.

Check the use of forbidden mathematical functions (see the subject).

Euclidean norm

Check the behaviour of the function with the following parameter:

  • '[0]' returns '0'.
  • '[1]' returns '1'.
  • '[0, 0]' returns '0'.
  • '[1, 0]' returns '1'.
  • '[2, 1]' returns '2.236067977'.
  • '[4, 2]' returns '4.472135955'.
  • '[-4, -2]' returns '4.472135955'.

Feel free to perform more tests on your own.

Manhattan norm

  • '[0]' returns '0'.
  • '[1]' returns '1'.
  • '[0, 0]' returns '0'.
  • '[1, 0]' returns '1'.
  • '[2, 1]' returns '3'.
  • '[4, 2]' returns '6'.
  • '[-4, -2]' returns '6'.

Feel free to perform more tests on your own.

Supremum norm

Test the function with several different vectors. Each time, the function
must return the component of the vector with the greatest value.

Exercise 05 - Cosine

Complexity

Ask the student to justify the complexity of the function. It must be at
most O(n) in time and O(n) in space.

Check the use of forbidden mathematical functions (see the subject).

Basic tests

Check the behaviour of the function with the following parameters:

  • '[1 0]' and '[0 1]' gives '0'
  • '[8 7]' and '[3 2]' gives '0.9914542955425437'
  • '[1 1]' and '[1 1]' gives '1'
  • '[4 2]' and '[1 1]' gives '0.9486832980505138'
  • '[-7 3]' and '[6 4]' gives '-0.5462677805469223'

Since the order of the parameters doesn't matter (the function is said to be
commutative), the function must return the same result if you swap them.

Feel free to perform more tests on your own.

Exercise 06 - Cross product

Basic tests

Check the behaviour of the function with the following parameters:

  • '[0 0 0]' and '[0 0 0]' gives '[0 0 0]'
  • '[1 0 0]' and '[0 0 0]' gives '[0 0 0]'
  • '[1 0 0]' and '[0 1 0]' gives '[0 0 1]'
  • '[8 7 -4]' and '[3 2 1]' gives '[15 -20 -5]'
  • '[1 1 1]' and '[0 0 0]' gives '[0 0 0]'
  • '[1 1 1]' and '[1 1 1]' gives '[0 0 0]'

Feel free to perform more tests on your own. When giving two vectors to
the function, imagine them creating a plane. Then, the function must return
a vector that is orthogonal (perpendicular) to that plane.

Check the use of forbidden mathematical functions (see the subject).

Exercise 07 - Linear transform

Complexity

Ask the student to justify the complexity of the function. It must be at
most O(n^3) in time and O(n^2) in space.

Check the use of forbidden mathematical functions (see the subject).

Basic tests

Check the behaviour of the function with the following parameter:

  • '[[0, 0], [0, 0]]' and any vector of dimension two. The function must always return vectors with only zeros in it.
  • '[[1, 0], [0, 1]]' and any vector of dimension two. The function must always return the same vector as given in parameter.
  • '[[1, 1], [1, 1]]' and '[4, 2]'. The function must return '[6, 6]'.
  • '[[2, 0], [0, 2]]' and '[2, 1]'. The function must return '[4, 2]'.
  • '[[0.5, 0], [0, 0.5]]' and '[4, 2]'. The function must return '[2, 1]'.

Feel free to perform more tests on your own

Exercise 08 - Trace

Complexity

Ask the student to justify the complexity of the function. It must be at most O(n) in time.

Check the use of forbidden mathematical functions (see the subject).

Basic tests

Check the behaviour of the function with the following parameter:

  • '[[0, 0], [0, 0]]' returns '0'
  • '[[1, 0], [0, 1]]' returns '2'
  • '[[1, 2], [3, 4]]' returns '5'
  • '[[8, -7], [4, 2]]' returns '10'
  • '[[1, 0, 0], [0, 1, 0], [0, 0, 1]]' returns '3'

Feel free to perform more tests on your own

Exercise 09 - Transpose

Complexity

Ask the student to justify the complexity of the function. It must be at
most O(n^2) (value assignments) in time and O(n^2) in space.

Check the use of forbidden mathematical functions (see the subject).

Basic tests

Check the behaviour of the function with the following parameter:

  • '[[0, 0], [0, 0]]' returns '[[0, 0], [0, 0]]'
  • '[[1, 0], [0, 1]]' returns '[[1, 0], [0, 1]]'
  • '[[1, 2], [3, 4]]' returns '[[1, 3], [2, 4]]'
  • '[[1, 0, 0], [0, 1, 0], [0, 0, 1]]' returns '[[1, 0, 0], [0, 1, 0], [0, 0, 1]]'
  • '[[1, 2], [3, 4], [5, 6]]' returns '[[1, 3, 5], [2, 4, 6]]'

Feel free to perform more tests on your own

Exercise 10 - row-echelon form

Complexity

Ask the student to justify the complexity of the function. It must be at
most O(n^3) in time and O(n^2) in space.

Check the use of forbidden mathematical functions (see the subject).

Basic tests

Check the behaviour of the function with the following parameter:

  • '[[0, 0], [0, 0]]' gives '[[0, 0], [0, 0]]'
  • '[[1, 0], [0, 1]]' gives '[[1, 0], [0, 1]]'
  • '[[4, 2], [2, 1]]' gives '[[1, 0.5], [0, 0]]'
  • '[[-7, 2], [4, 8]]' gives '[[1, 0], [0, 1]]'
  • '[[1, 2], [4, 8]]' gives '[[1, 2], [0, 0]]'

Feel free to perform more tests on your own

Exercise 11 - Determinant

Complexity

Ask the student to justify the complexity of the function. It must be at
most O(n^3) in time.

Check the use of forbidden mathematical functions (see the subject).

Basic tests

Check the behaviour of the function with the following parameter:

  • '[[0, 0], [0, 0]]' returns '0'
  • '[[1, 0], [0, 1]]' returns '1'
  • '[[2, 0], [0, 2]]' returns '4'
  • '[[1, 1], [1, 1]]' returns '0'
  • '[[0, 1], [1, 0]]' returns '-1'
  • '[[1, 2], [3, 4]]' returns '-2'
  • '[[-7, 5], [4, 6]]' returns '-62'
  • '[[1, 0, 0], [0, 1, 0], [0, 0, 1]]' returns '1'

Feel free to perform more tests on your own

Explanations

Ask the student to explain:

  • What happens when the determinant of a matrix is '0'.
  • What the determinant represents geometrically in the vector space (ie, what happens after using the matrix for a linear transformation, and what does the determinant describe)

If they cannot explain it, the evaluation ends here.

Exercise 12 - Inverse

Complexity

Ask the student to justify the complexity of the function. It must be at
most O(n^3) in time and O(n^2) in space.

Check the use of forbidden mathematical functions (see the subject).

Basic tests

Check the behaviour of the function with the following parameter:

  • '[[1, 0], [0, 1]]' returns '[[1, 0], [0, 1]]'
  • '[[2, 0], [0, 2]]' returns '[[0.5, 0], [0, 0.5]]'
  • '[[0.5, 0], [0, 0.5]]' returns '[[2, 0], [0, 2]]'
  • '[[0, 1], [1, 0]]' returns '[[0, 1], [1, 0]]'
  • '[[1, 2], [3, 4]]' returns '[[-2, 1], [1.5, -0.5]]'
  • '[[1, 0, 0], [0, 1, 0], [0, 0, 1]]' returns '[[1, 0, 0], [0, 1, 0], [0, 0, 1]]'

Feel free to perform more tests on your own. To check the result, you can
multiply it by the matrix you gave as parameter and it must give (approximately)
the identity matrix (However, avoid testing matrices that are not invertible).

Exercise 13 - Rank

Basic tests

Check the behaviour of the function with the following parameter:

  • '[[0, 0], [0, 0]]' returns '0'
  • '[[1, 0], [0, 1]]' returns '2'
  • '[[2, 0], [0, 2]]' returns '2'
  • '[[1, 1], [1, 1]]' returns '1'
  • '[[0, 1], [1, 0]]' returns '2'
  • '[[1, 2], [3, 4]]' returns '2'
  • '[[-7, 5], [4, 6]]' returns '2'
  • '[[1, 0, 0], [0, 1, 0], [0, 0, 1]]' returns '3'

Feel free to perform more tests on your own

Check the use of forbidden mathematical functions (see the subject).

Explanations

Ask the student to explain what the rank of a matrix represents.

If they cannot explain it, the evaluation ends here. You can use the
internet to check the answers.

Exercise 14 - Bonus: Projection matrix

Projection

Build several matrices with several FoVs (convert the value in radians before
passing it to the function):

  • 100 degrees
  • 70 degrees
  • 40 degrees

Then, test the matrices in the projection utility given in the attachements.

Also, try testing with several different combinations of near/far values (near
must stay smaller than far) and different ratios (the default is 1).

A lower FoV must reduce the angle of view.

Changing the ratio must distort the image.

Different values of near and far must change the distance from the camera at
which objects disappear from the screen.

Ask the student to explain what each component of the matrix represents.

Exercise 15 - Bonus: Complex vector spaces

Lots of tests

For this exercise, the student must have recoded all the previous functions (except for ex14),
or used the generic structure of the code, to provide the use of complex
numbers as scalars. The student should be able to explain how the operations
of complex numbers work (geometrically).

Reminder of the rules for complex numbers:

  • 'i^2 = -1'
  • '(a + bi) + (c + di) = (a + c) + (b + d)i'
  • '(a + bi) - (c + di) = (a - c) + (b - d)i'
  • '(a + bi) * (c + di) = (ac - bd) + (bc + ad)i'
  • '(a + bi) / (c + di) = ((ac + bd) + (bc - ad)i) / (c^2 + d^2)'

Test every function, but with complex numbers, and check that they behave
correctly. The student that has done this bonus should probably provide
tests for complex numbers in his executables, and show them along with
the correction for the regular exercises, if they wish to gain time.

Ratings

Don’t forget to check the flag corresponding to the defense

Conclusion

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