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| --- | ||
| author: MathWorks Blogs | ||
| author_tag: mathworks | ||
| blog_subtitle: Cleve Moler is the author of the first MATLAB, one of the founders | ||
| of MathWorks, and is currently Chief Mathematician at the company. He writes here | ||
| about MATLAB, scientific computing and interesting mathematics. | ||
| blog_title: 'Cleve’s Corner: Cleve Moler on Mathematics and Computing' | ||
| blog_url: https://blogs.mathworks.com/cleve | ||
| category: mathworks | ||
| date: '2023-09-25 17:57:28' | ||
| layout: post | ||
| original_url: https://blogs.mathworks.com/cleve/2023/09/25/exploring-matrices-exercises/?s_tid=feedtopost | ||
| slug: exploring-matrices-exercises | ||
| title: Exploring Matrices Exercises | ||
| --- | ||
|
|
||
| <div class="content"><!--introduction--><p>Try your hand at a few exercises involving <a href="https://blogs.mathworks.com/cleve/2023/09/11/exploring-matrices">Exploring Matrices</a>.</p> | ||
| <!--/introduction--><h3>Contents</h3><div><ul><li><a href="https://feeds.feedburner.com/mathworks/moler#39c51606-0f1d-4f63-a836-a5b7b53bd13b">Qube Simplified</a></li><li><a href="https://feeds.feedburner.com/mathworks/moler#7c52cde6-70d3-4ad5-b1a1-37c5ce6e87cc">Exploring Matrices Exercises</a></li><li><a href="https://feeds.feedburner.com/mathworks/moler#72d1baf3-32a5-43fd-9ee9-d78b0756d365">Matrix Multiplication</a></li><li><a href="https://feeds.feedburner.com/mathworks/moler#5866120b-8ad5-44c6-91d4-7599fdd930ad">Rotations and Scaling</a></li><li><a href="https://feeds.feedburner.com/mathworks/moler#9ee4efdc-6cce-4067-b642-b70c3c75d2c6">Computer Graphics</a></li><li><a href="https://feeds.feedburner.com/mathworks/moler#08826629-b0f9-4a97-a3e3-655f52742c85">Matrices and Cubes</a></li></ul></div> | ||
| <h4>Qube Simplified<a name="39c51606-0f1d-4f63-a836-a5b7b53bd13b"></a></h4><p><img alt="" hspace="5" src="https://blogs.mathworks.com/cleve/files/Qube_simplified.png" vspace="5" /> </p> | ||
| <p>I have simplified the <tt>Qube</tt> app by removing these four buttons.</p> | ||
| <div><ul><li><tt>solve</tt>. The <== key now controls the unscrambling operation.</li></ul></div> | ||
| <div><ul><li><tt>scramble</tt>. The ==> key now does six random rotations.</li></ul></div> | ||
| <div><ul><li><tt>order</tt>. I never found a satisfactory reference for the group theory of Rubik's cube.</li></ul></div> | ||
| <div><ul><li><tt>score</tt>. I never found a use for the nuclear norm.</li></ul></div> | ||
| <p>Code for <tt>Qube</tt> dated 9/24/2023 is included in the <a href="https://blogs.mathworks.com/cleve/files/Apps_mzip.m">Apps mzip archive</a>.</p> | ||
| <h4>Exploring Matrices Exercises<a name="7c52cde6-70d3-4ad5-b1a1-37c5ce6e87cc"></a></h4><p>Here are a few exercises for <a href="https://blogs.mathworks.com/cleve/2023/09/11/exploring-matrices">Exploring Matrices</a>. The answers are available at <a href="https://blogs.mathworks.com/cleve/files/ExMatAnswers.pdf">ExMatAnswers</a>.</p> | ||
| <h4>Matrix Multiplication<a name="72d1baf3-32a5-43fd-9ee9-d78b0756d365"></a></h4><p><b>1.</b> Compute by rows, and by columns.</p> | ||
| <p>$$ | ||
| \left( | ||
| \begin{array}{rrr} | ||
| 8 & 1 & 6 \\ | ||
| 3 & 5 & 7 \\ | ||
| 4 & 9 & 2 | ||
| \end{array} | ||
| \right) | ||
| \left( | ||
| \begin{array}{r} | ||
| 1 \\ | ||
| 1 \\ | ||
| 1 | ||
| \end{array} | ||
| \right) | ||
| $$</p> | ||
| <p><b>2.</b> Solve for $z$ using inner products of rows, and using linear combinations of columns.</p> | ||
| <p>$$ | ||
| \left( | ||
| \begin{array}{rrr} | ||
| 1 & 2 & 3 \\ | ||
| 4 & 5 & 6 \\ | ||
| 7 & 8 & 9 | ||
| \end{array} | ||
| \right) | ||
| \left( | ||
| \begin{array}{r} | ||
| 1 \\ | ||
| z \\ | ||
| 1 | ||
| \end{array} | ||
| \right) | ||
| \ = \ | ||
| \left( | ||
| \begin{array}{r} | ||
| 0 \\ | ||
| 0 \\ | ||
| 0 | ||
| \end{array} | ||
| \right) | ||
| $$</p> | ||
| <p><b>3.</b> What do the <tt>m</tt>, <tt>n</tt> and <tt>p</tt> buttons on the <tt>Multiply</tt> app do? What are the other buttons and what do they do?</p> | ||
| <p><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> ?</p> | ||
| <p><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> ?</p> | ||
| <h4>Rotations and Scaling<a name="5866120b-8ad5-44c6-91d4-7599fdd930ad"></a></h4><p><b>1.</b> What is <i>R</i>(30º)?</p> | ||
| <p>$$ | ||
| R(\theta) \ = \ | ||
| \left( | ||
| \begin{array}{rr} | ||
| \cos{\theta} & \sin{\theta} \\ | ||
| -\sin{\theta} & \cos{\theta} | ||
| \end{array} | ||
| \right) | ||
| $$</p> | ||
| <p><b>2.</b> Explain <a href="https://xkcd.com/184">https://xkcd.com/184</a>.</p> | ||
| <p><b>3.</b> What is the value of $\theta$ ?</p> | ||
| <p>$$ | ||
| R(\theta) \ = \ | ||
| \left( | ||
| \begin{array}{rr} | ||
| 0.8 & 0.6 \\ | ||
| -0.6 & 0.8 | ||
| \end{array} | ||
| \right) | ||
| $$</p> | ||
| <p><b>4.</b> What is the value of $\theta$ ?</p> | ||
| <p><img alt="" hspace="5" src="https://blogs.mathworks.com/cleve/files/theta_exer.png" vspace="5" /> </p> | ||
| <p><b>5.</b> Edit a copy of <tt>Rotate.m</tt> and replace the house with a hand. You can use <a href="https://mathworks-my.sharepoint.com/:u:/p/moler/EVREwZQo4ZBLkDggPTky6o4BNIfckvaLQoCDdX4ICjch9A?e=dGA2Y4">my hand</a> or your own hand; see exercise 3.4 in <a href="https://www.mathworks.com/content/dam/mathworks/mathworks-dot-com/moler/interp.pdf">Numerical Computing with MATLAB</a> .</p> | ||
| <p><img alt="" hspace="5" src="https://blogs.mathworks.com/cleve/files/hand_exer.png" vspace="5" /> </p> | ||
| <h4>Computer Graphics<a name="9ee4efdc-6cce-4067-b642-b70c3c75d2c6"></a></h4><p><b>1.</b> Show how homogeneous coordinates and matrix-vector multiplication by <tt>Tx</tt>, <tt>Ty</tt> or <tt>Tz</tt> produce translation.</p> | ||
| <p><b>2.</b> What is the range of the rotations used by the pitch, roll, and yaw buttons on the <tt>Grafix</tt> app?</p> | ||
| <p><b>3.</b> What color in the beacon on top of the plane? How would you change the beacon's color?</p> | ||
| <p><b>4,</b> What is the function of the resolution and offset sliders for the teapot?</p> | ||
| <p><b>5,</b> How many times does the bucky ball bounce off the sides of the plot window?</p> | ||
| <h4>Matrices and Cubes<a name="08826629-b0f9-4a97-a3e3-655f52742c85"></a></h4><p><b>1.</b> What is the color of central cubelet in the Color Cube?</p> | ||
| <p><b>2.</b> What do the "<=" , "<==" , "=>" and "==>" buttons on <tt>Qube</tt> do?</p> | ||
| <p><b>3.</b> What is "God's Number" for a 3-by-3-by-3 Rubik's Cube? What are <tt>Q20</tt> and <tt>Q26</tt>? See <a href="https://blogs.mathworks.com/cleve/2022/09/05/rubiks-cube-superflips-and-gods-number">Cleve's Corner 2022/09/05</a>.</p> | ||
| <p><b>4.</b> Can you restore the following scrambled cubes with fewer moves than <tt><==</tt>, the unscramble key? Use the quarter-turn metric and reset the cube with <tt>start</tt> or by clicking on <tt>stack</tt> and <tt>Q0</tt>. You might also want to set <tt>speed</tt> to 30 or 45,</p> | ||
| <div><ul><li><tt>LRUDFB</tt></li></ul></div> | ||
| <div><ul><li><tt>LRL'R'</tt></li></ul></div> | ||
| <div><ul><li><tt>FLLLRB</tt></li></ul></div> | ||
| <div><ul><li><tt>Q26</tt></li></ul></div> | ||
| <div><ul><li>Reset the random number generator by entering <tt>rng(r)</tt> for some small integer <tt>r</tt> in the command window and then generate six random rotations with the <tt>==></tt> key.</li></ul></div> | ||
| <!-- | ||
| function grabCode_661f86ea0d734810b8ca2514e6ba69c9() { | ||
| // Remember the title so we can use it in the new page | ||
| title = document.title; | ||
| // Break up these strings so that their presence | ||
| // in the Javascript doesn't mess up the search for | ||
| // the MATLAB code. | ||
| t1='661f86ea0d734810b8ca2514e6ba69c9 ' + '##### ' + 'SOURCE BEGIN' + ' #####'; | ||
| t2='##### ' + 'SOURCE END' + ' #####' + ' 661f86ea0d734810b8ca2514e6ba69c9'; | ||
| b=document.getElementsByTagName('body')[0]; | ||
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| i2=b.innerHTML.indexOf(t2); | ||
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| code_string = code_string.replace(/REPLACE_WITH_DASH_DASH/g,'--'); | ||
| // Use /x3C/g instead of the less-than character to avoid errors | ||
| // in the XML parser. | ||
| // Use '\x26#60;' instead of '<' so that the XML parser | ||
| // doesn't go ahead and substitute the less-than character. | ||
| code_string = code_string.replace(/\x3C/g, '\x26#60;'); | ||
| copyright = 'Copyright 2023 The MathWorks, Inc.'; | ||
| w = window.open(); | ||
| d = w.document; | ||
| d.write('<pre>\n'); | ||
| d.write(code_string); | ||
| // Add copyright line at the bottom if specified. | ||
| if (copyright.length > 0) { | ||
| d.writeln(''); | ||
| d.writeln('%%'); | ||
| if (copyright.length > 0) { | ||
| d.writeln('% _' + copyright + '_'); | ||
| } | ||
| } | ||
| d.write('</pre>\n'); | ||
| d.title = title + ' (MATLAB code)'; | ||
| d.close(); | ||
| } | ||
| --><p style="text-align: right; font-size: xx-small; font-weight: lighter; font-style: italic; color: gray;"><br /><a href=""><span style="font-size: x-small; font-style: italic;">Get | ||
| the MATLAB code <noscript>(requires JavaScript)</noscript></span></a><br /><br /> | ||
| Published with MATLAB® R2023a<br /></p> | ||
| </div> | ||
| <!-- | ||
| 661f86ea0d734810b8ca2514e6ba69c9 ##### SOURCE BEGIN ##### | ||
| %% Exploring Matrices Exercises | ||
| % Try your hand at a few exercises involving | ||
| % <https://blogs.mathworks.com/cleve/2023/09/11/exploring-matrices | ||
| % Exploring Matrices>. | ||
| %% Qube Simplified | ||
| % | ||
| % <<Qube_simplified.png>> | ||
| % | ||
| % I have simplified the |Qube| app by removing these four buttons. | ||
| % | ||
| % * |solve|. The <== key now controls the unscrambling | ||
| % operation. | ||
| % | ||
| % * |scramble|. The ==> key now does six random rotations. | ||
| % | ||
| % * |order|. I never found a satisfactory reference for | ||
| % the group theory of Rubik's cube. | ||
| % | ||
| % * |score|. I never found a use for the nuclear norm. | ||
| % | ||
| % Code for |Qube| dated 9/24/2023 is included in the | ||
| % <https://blogs.mathworks.com/cleve/files/Apps_mzip.m | ||
| % Apps mzip archive>. | ||
| %% Exploring Matrices Exercises | ||
| % | ||
| % Here are a few exercises for | ||
| % <https://blogs.mathworks.com/cleve/2023/09/11/exploring-matrices | ||
| % Exploring Matrices>. | ||
| % The answers are available at | ||
| % <https://blogs.mathworks.com/cleve/files/ExMatAnswers.pdf | ||
| % ExMatAnswers>. | ||
| %% Matrix Multiplication | ||
| % | ||
| % *1.* | ||
| % Compute by rows, and by columns. | ||
| % | ||
| % $$ | ||
| % \left( | ||
| % \begin{array}{rrr} | ||
| % 8 & 1 & 6 \\ | ||
| % 3 & 5 & 7 \\ | ||
| % 4 & 9 & 2 | ||
| % \end{array} | ||
| % \right) | ||
| % \left( | ||
| % \begin{array}{r} | ||
| % 1 \\ | ||
| % 1 \\ | ||
| % 1 | ||
| % \end{array} | ||
| % \right) | ||
| % $$ | ||
| % | ||
| % | ||
| % *2.* | ||
| % Solve for $z$ using inner products of rows, and | ||
| % using linear combinations of columns. | ||
| % | ||
| % $$ | ||
| % \left( | ||
| % \begin{array}{rrr} | ||
| % 1 & 2 & 3 \\ | ||
| % 4 & 5 & 6 \\ | ||
| % 7 & 8 & 9 | ||
| % \end{array} | ||
| % \right) | ||
| % \left( | ||
| % \begin{array}{r} | ||
| % 1 \\ | ||
| % z \\ | ||
| % 1 | ||
| % \end{array} | ||
| % \right) | ||
| % \ = \ | ||
| % \left( | ||
| % \begin{array}{r} | ||
| % 0 \\ | ||
| % 0 \\ | ||
| % 0 | ||
| % \end{array} | ||
| % \right) | ||
| % $$ | ||
| % | ||
| % | ||
| % *3.* | ||
| % What do the |m|, |n| and |p| buttons on the |Multiply| app do? | ||
| % What are the other buttons and what do they do? | ||
| % | ||
| % | ||
| % *4.* | ||
| % If _A_ is n-by-n and _x_ is n-by-1, | ||
| % how many multiplications are required to compute _A x_ ? | ||
| % | ||
| % *5.* | ||
| % If _A_ is m-by-n and _B_ is n-by-p, | ||
| % how many multiplications are required to compute _A B_ ? | ||
|
|
||
| %% Rotations and Scaling | ||
| % | ||
| % *1.* | ||
| % What is _R_(30º)? | ||
| % | ||
| % $$ | ||
| % R(\theta) \ = \ | ||
| % \left( | ||
| % \begin{array}{rr} | ||
| % \cos{\theta} & \sin{\theta} \\ | ||
| % -\sin{\theta} & \cos{\theta} | ||
| % \end{array} | ||
| % \right) | ||
| % $$ | ||
| % | ||
| % *2.* | ||
| % Explain <https://xkcd.com/184>. | ||
| % | ||
| % *3.* | ||
| % What is the value of $\theta$ ? | ||
| % | ||
| % $$ | ||
| % R(\theta) \ = \ | ||
| % \left( | ||
| % \begin{array}{rr} | ||
| % 0.8 & 0.6 \\ | ||
| % -0.6 & 0.8 | ||
| % \end{array} | ||
| % \right) | ||
| % $$ | ||
| % | ||
| % | ||
| % *4.* | ||
| % What is the value of $\theta$ ? | ||
| % | ||
| % <<theta_exer.png>> | ||
| % | ||
| % | ||
| % *5.* | ||
| % Edit a copy of |Rotate.m| and replace the house with a hand. You can use | ||
| % <https://mathworks-my.sharepoint.com/:u:/p/moler/EVREwZQo4ZBLkDggPTky6o4BNIfckvaLQoCDdX4ICjch9A?e=dGA2Y4 my hand> | ||
| % or your own hand; see exercise 3.4 in | ||
| % <https://www.mathworks.com/content/dam/mathworks/mathworks-dot-com/moler/interp.pdf | ||
| % Numerical Computing with MATLAB> . | ||
| % | ||
| % <<hand_exer.png>> | ||
| % | ||
|
|
||
| %% Computer Graphics | ||
| % | ||
| % *1.* | ||
| % Show how homogeneous coordinates and matrix-vector multiplication by | ||
| % |Tx|, |Ty| or |Tz| produce translation. | ||
| % | ||
| % | ||
| % *2.* | ||
| % What is the range of the rotations used by the pitch, roll, and yaw | ||
| % buttons on the |Grafix| app? | ||
| % | ||
| % *3.* | ||
| % What color in the beacon on top of the plane? | ||
| % How would you change the beacon's color? | ||
| % | ||
| % | ||
| % *4,* | ||
| % What is the function of the resolution and offset sliders for the teapot? | ||
| % | ||
| % | ||
| % *5,* | ||
| % How many times does the bucky ball bounce off the sides of the plot | ||
| % window? | ||
| % | ||
|
|
||
|
|
||
| %% Matrices and Cubes | ||
| % | ||
| % *1.* | ||
| % What is the color of central cubelet in the Color Cube? | ||
| % | ||
| % | ||
| % *2.* | ||
| % What do the "<=" , "<==" , "=>" and "==>" buttons on |Qube| do? | ||
| % | ||
| % | ||
| % *3.* | ||
| % What is "God's Number" for a 3-by-3-by-3 Rubik's Cube? | ||
| % What are |Q20| and |Q26|? | ||
| % See <https://blogs.mathworks.com/cleve/2022/09/05/rubiks-cube-superflips-and-gods-number | ||
| % Cleve's Corner 2022/09/05>. | ||
| % | ||
| % | ||
| % *4.* | ||
| % Can you restore the following scrambled cubes with fewer moves than |<==|, | ||
| % the unscramble key? Use the quarter-turn metric and | ||
| % reset the cube with |start| or by clicking on |stack| and |Q0|. | ||
| % You might also want to set |speed| to 30 or 45, | ||
| % | ||
| % * |LRUDFB| | ||
| % | ||
| % * |LRL'R'| | ||
| % | ||
| % * |FLLLRB| | ||
| % | ||
| % * |Q26| | ||
| % | ||
| % * Reset the random number generator by entering |rng(r)| for some | ||
| % small integer |r| in the command window and then generate six | ||
| % random rotations with the |==>| key. | ||
|
|
||
|
|
||
| ##### SOURCE END ##### 661f86ea0d734810b8ca2514e6ba69c9 | ||
| --> |