Attempt your hand at a number of workout routines involving Exploring Matrices.
Contents
Qube Simplified
I’ve simplified the Qube app by eradicating these 4 buttons.
- resolve. The <== key now controls the unscrambling operation.
- scramble. The ==> key now does six random rotations.
- order. I by no means discovered a passable reference for the group idea of Rubik’s dice.
- rating. I by no means discovered a use for the nuclear norm.
Code for Qube dated 9/24/2023 is included within the Apps mzip archive.
Exploring Matrices Workouts
Listed here are a number of workout routines for Exploring Matrices. The solutions can be found at ExMatAnswers.
Matrix Multiplication
1. Compute by rows, and by columns.
$$
left(
start{array}{rrr}
8 & 1 & 6
3 & 5 & 7
4 & 9 & 2
finish{array}
proper)
left(
start{array}{r}
1
1
1
finish{array}
proper)
$$
2. Remedy for $z$ utilizing internal merchandise of rows, and utilizing linear combos of columns.
$$
left(
start{array}{rrr}
1 & 2 & 3
4 & 5 & 6
7 & 8 & 9
finish{array}
proper)
left(
start{array}{r}
1
z
1
finish{array}
proper)
=
left(
start{array}{r}
0
0
0
finish{array}
proper)
$$
3. What do the m, n and p buttons on the Multiply app do? What are the opposite buttons and what do they do?
4. If A is n-by-n and x is n-by-1, what number of multiplications are required to compute A x ?
5. If A is m-by-n and B is n-by-p, what number of multiplications are required to compute A B ?
Rotations and Scaling
1. What’s R(30º)?
$$
R(theta) =
left(
start{array}{rr}
cos{theta} & sin{theta}
-sin{theta} & cos{theta}
finish{array}
proper)
$$
2. Clarify https://xkcd.com/184.
3. What’s the worth of $theta$ ?
$$
R(theta) =
left(
start{array}{rr}
0.8 & 0.6
-0.6 & 0.8
finish{array}
proper)
$$
4. What’s the worth of $theta$ ?
5. Edit a duplicate of Rotate.m and substitute the home with a hand. You should use my hand or your individual hand; see train 3.4 in Numerical Computing with MATLAB .
Pc Graphics
1. Present how homogeneous coordinates and matrix-vector multiplication by Tx, Ty or Tz produce translation.
2. What’s the vary of the rotations utilized by the pitch, roll, and yaw buttons on the Grafix app?
3. What shade within the beacon on high of the aircraft? How would you alter the beacon’s shade?
4, What’s the perform of the decision and offset sliders for the teapot?
5, What number of occasions does the bucky ball bounce off the edges of the plot window?
Matrices and Cubes
1. What’s the shade of central cubelet within the Coloration Dice?
2. What do the “<=” , “<==” , “=>” and “==>” buttons on Qube do?
3. What’s “God’s Quantity” for a 3-by-3-by-3 Rubik’s Dice? What are Q20 and Q26? See Cleve’s Nook 2022/09/05.
4. Are you able to restore the next scrambled cubes with fewer strikes than <==, the unscramble key? Use the quarter-turn metric and reset the dice with begin or by clicking on stack and Q0. You may also wish to set velocity to 30 or 45,
- Reset the random quantity generator by getting into rng(r) for some small integer r within the command window after which generate six random rotations with the ==> key.
Printed with MATLAB® R2023a
