Exploring Matrices Exercises - Cleve Moler on Mathematics and Computing
 

Try your hand at a few exercises involving [URL="https://blogs.mathworks.com/cleve/2023/09/11/exploring-matrices"]Exploring Matrices[/URL].

[B]Contents[/B]
[LIST][*][URL="https://www.labfans.com/bbs/#39c51606-0f1d-4f63-a836-a5b7b53bd13b"]Qube Simplified[/URL][*][URL="https://www.labfans.com/bbs/#7c52cde6-70d3-4ad5-b1a1-37c5ce6e87cc"]Exploring Matrices Exercises[/URL][*][URL="https://www.labfans.com/bbs/#72d1baf3-32a5-43fd-9ee9-d78b0756d365"]Matrix Multiplication[/URL][*][URL="https://www.labfans.com/bbs/#5866120b-8ad5-44c6-91d4-7599fdd930ad"]Rotations and Scaling[/URL][*][URL="https://www.labfans.com/bbs/#9ee4efdc-6cce-4067-b642-b70c3c75d2c6"]Computer Graphics[/URL][*][URL="https://www.labfans.com/bbs/#08826629-b0f9-4a97-a3e3-655f52742c85"]Matrices and Cubes[/URL][/LIST][B]Qube Simplified[/B]

[IMG]https://blogs.mathworks.com/cleve/files/Qube_simplified.png[/IMG]

I have simplified the Qube app by removing these four buttons.
[LIST][*]solve. The key now does six random rotations.[/LIST][LIST][*]order. I never found a satisfactory reference for the group theory of Rubik's cube.[/LIST][LIST][*]score. I never found a use for the nuclear norm.[/LIST]Code for Qube dated 9/24/2023 is included in the [URL="https://blogs.mathworks.com/cleve/files/Apps_mzip.m"]Apps mzip archive[/URL].

[B]Exploring Matrices Exercises[/B]

Here are a few exercises for [URL="https://blogs.mathworks.com/cleve/2023/09/11/exploring-matrices"]Exploring Matrices[/URL]. The answers are available at [URL="https://blogs.mathworks.com/cleve/files/ExMatAnswers.pdf"]ExMatAnswers[/URL].

[B]Matrix Multiplication[/B]

[B]1.[/B] Compute by rows, and by columns.

$$ \left( \begin{array}{rrr} 8 & 1 & 6 \\ 3 & 5 & 7 \\ 4 & 9 & 2 \end{array} \right) \left( \begin{array}{r} 1 \\ 1 \\ 1 \end{array} \right)$$

[B]2.[/B] Solve for $z$ using inner products of rows, and using linear combinations of columns.

$$ \left( \begin{array}{rrr} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array} \right) \left( \begin{array}{r} 1 \\ z \\ 1 \end{array} \right) \ = \ \left( \begin{array}{r} 0 \\ 0 \\ 0 \end{array} \right)$$

[B]3.[/B] What do the m, n and p buttons on the Multiply app do? What are the other buttons and what do they do?

[B]4.[/B] If [I]A[/I] is n-by-n and [I]x[/I] is n-by-1, how many multiplications are required to compute [I]A x[/I] ?

[B]5.[/B] If [I]A[/I] is m-by-n and [I]B[/I] is n-by-p, how many multiplications are required to compute [I]A B[/I] ?

[B]Rotations and Scaling[/B]

[B]1.[/B] What is [I]R[/I](30º)?

$$R(\theta) \ = \\left( \begin{array}{rr} \cos{\theta} & \sin{\theta} \\ -\sin{\theta} & \cos{\theta} \end{array} \right)$$

[B]2.[/B] Explain [URL]https://xkcd.com/184[/URL].

[B]3.[/B] What is the value of $\theta$ ?

$$R(\theta) \ = \\left( \begin{array}{rr} 0.8 & 0.6 \\ -0.6 & 0.8 \end{array} \right)$$

[B]4.[/B] What is the value of $\theta$ ?

[IMG]https://blogs.mathworks.com/cleve/files/theta_exer.png[/IMG]

[B]5.[/B] Edit a copy of Rotate.m and replace the house with a hand. You can use [URL="https://mathworks-my.sharepoint.com/:u:/p/moler/EVREwZQo4ZBLkDggPTky6o4BNIfckvaLQoCDdX4ICjch9A?e=dGA2Y4"]my hand[/URL] or your own hand; see exercise 3.4 in [URL="https://www.mathworks.com/content/dam/mathworks/mathworks-dot-com/moler/interp.pdf"]Numerical Computing with MATLAB[/URL] .

[IMG]https://blogs.mathworks.com/cleve/files/hand_exer.png[/IMG]

[B]Computer Graphics[/B]

[B]1.[/B] Show how homogeneous coordinates and matrix-vector multiplication by Tx, Ty or Tz produce translation.

[B]2.[/B] What is the range of the rotations used by the pitch, roll, and yaw buttons on the Grafix app?

[B]3.[/B] What color in the beacon on top of the plane? How would you change the beacon's color?

[B]4,[/B] What is the function of the resolution and offset sliders for the teapot?

[B]5,[/B] How many times does the bucky ball bounce off the sides of the plot window?

[B]Matrices and Cubes[/B]

[B]1.[/B] What is the color of central cubelet in the Color Cube?

[B]2.[/B] What do the "" buttons on Qube do?

[B]3.[/B] What is "God's Number" for a 3-by-3-by-3 Rubik's Cube? What are Q20 and Q26? See [URL="https://blogs.mathworks.com/cleve/2022/09/05/rubiks-cube-superflips-and-gods-number"]Cleve's Corner 2022/09/05[/URL].


[B]4.[/B] Can you restore the following scrambled cubes with fewer moves than key.</ul>
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[URL="javascript:grabCode_661f86ea0d734810b8ca2514e6ba69c9()"][I]Get the MATLAB code (requires JavaScript)[/I][/URL]

Published with MATLAB&reg; R2023a
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