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MAST10007 Linear Algebra

创建日期：2024-06-15 15:44:22

所属分类：数学mathematics

标签： MAST10007

Linear Algebra

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MAST10007 Linear Algebra

Task 1: Convince yourself that T is a regular tetrahedron. Do do so, you need to compute some
distances and angles, for instance
• the distance between a⃗ and b⃗,
• the distance between a⃗ and d⃗,
• the length of the vectors a⃗, b⃗, c⃗, and d⃗,
• the angle between b⃗− a⃗ and c⃗− a⃗,
• the angle between b⃗− a⃗ and d⃗− a⃗,
• the angle between a⃗− d⃗ and b⃗− d⃗, and so on.
• you can also compute the sum of these four vectors, if you like.
Compute just enough of these so you practice distances and angles and you are confident in the end that T
is a mathematical description of the object you are meant to craft below.
Next,
• Write
10
0
 as a linear combination of a⃗ and d⃗.
• Write
01
0
 as a linear combination of b⃗ and d⃗.
• Write
00
1
 as a linear combination of c⃗ and d⃗.
Task 3: Find the symmetries and fill in the composition table. You are now in a position to find
the matrices of all the linear symmetries of the tetrahedron.
The first rotation Start with the symmetry A→ B → C → A. You are looking for a 3× 3-matrix R, and
you know that R needs to satisfy
Ra⃗ = b⃗ and Rb⃗ = c⃗ and Rc⃗ = a⃗ and Rd⃗ = d⃗.
Use this information to find
R
10
0
 and R
01
0
 and R
00
1

and hence R. Make sure to fact check your computation by calculating Ra⃗ and Rb⃗ and Rc⃗ and Rd⃗.
Right-hand rule Place a piece of paper on your desk and place the tetrahedron you crafted on top with
the face D facing down and the vertex d⃗ at the top. On your piece of paper, mark and label the positions
of the other three vertices. Your rotation moves the vertex a⃗ to position b⃗, the vertex b⃗ to position c⃗ and
the vertex c⃗ to position a⃗. Move your tetrahedron accordingly. Put your right fist on the desk next to your
tetrahedron, thumb facing towards the ceiling. So, your thumb is facing in the same direction as the vector
d⃗. Open your fist slightly, and turn rotate your hand, in the direction of your fingertips. Your thumb should
continue to face up. This is the positive direction of rotation, according to the right-hand rule, about the
vector d⃗. You can figure out the angle and its sign from how you moved your tetrahedron.
The second rotation Next, find the matrix of the symmetry A↔ B, C ↔ D. Let’s call this matrix S,
we will need it later. Use the same method as you used to find R. You know what S does to the vertices of
T . Use this knoweldge to determine the effect of S on the standard basis vectors. This requires a little bit
of patience. The computation is easier if you always immediately pull the common denominator out of your
vector or matrix.
Take your tetrahedron. Can you figure out the rotation axis for S? Without any computations, can you
find S2? What is the rotation angle?
1
Fact-check your answers in multiple ways: Form Sa⃗, Sb⃗, Sc⃗, and Sd⃗. You can also compute the length
of each column vector of S (this should be 1, why?) and their dot product with each other (this should
be 0 unless the vectors are equal to each other, why?). Multiply S with the vector you believe spans the
rotation axis. Does multiplication by S leave this vector unchanged? Compute S2 and compare it with your
geometric prediction. Use MATLAB for these fact-checks, if you get tired of doing them by hand. You want
to be absolutely certain that you found the correct matrices for R and S, because you will use these to find
the others.
Finding all the other rotations From here, you have a number of options on how to proceed. A
tedious approach would be to repeat the method above for each of the remaining nine symmetries. An easier
way is to use the fact that a symmetry followed by another symmetry is again a symmetry. Composition of
symmetries corresponds to matrix multiplication. So, start by computing some matrix products, using the
two matrices R and S that you have already found. For instance, compute
R2, RS, SR, R2S, RSR, SR2
and so on. Always go for the easiest product that you have not already computed. Once you have computed
a new symmetry in this manner, figure out what is does to the vertices of T and enter the answer in the
corresponding box on the answer sheet. It is important that you bring the answers to paper in the prescribed
order, so your tutors find the matrix where they are looking for it. (Gradescope takes forever to turn pages.)
You can fact-check your answers and keep track of your computations using the composition table.
How to use the composition table: you probably do not want to calculate 144 matrix products, but
you can work out relatively easily how the symmetries compose: for each box, the symmetry labeling its
column is carried out first, followed by the symmetry labeling its row.
In the example that is filled in already, you have to first carry out the symmetry swapping face A with
face B and face C with face D, and then follow it by the symmetry that sends the face A to B, B to C and
C to A. This corresponds to the matrix product RS.
To use your tetrahedron to determine this symmetry, place it in its original position on your sheet of
paper. First apply the symmetry encoded in S. Afterwards the vertex c⃗ is on top (i.e., it has moved to the
position d⃗), while the vertex a⃗ is in position b⃗, the vertex b⃗ is in position a⃗, and the vertex d⃗ is in position
c⃗. The next symmetry is R. It gets applied to the positions. So, it is the same movement as before, keeping
the vertex c⃗ fixed at the top and rotating the tetrahedron about this position.
So, when you carry out both of these symmetries in order, A first gets send to B and from there to C,
the face B gets send to A and then back to B, the face C is send to D and the face D is sent to C and then
on to A. This is recorded in the composition table.

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