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===== Example Wrap-up Discussion #2===== | ===== Example Wrap-up Discussion #2===== | ||

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* Day 1 had been a 50-minute interactive lecture that introduced basic matrix manipulations such as matrix multiplication and determinants. | * Day 1 had been a 50-minute interactive lecture that introduced basic matrix manipulations such as matrix multiplication and determinants. | ||

- | * Day 2 had been a double period. First the students worked in small groups. Each group used a given matrix to transform a set of given vectors and then plotted the transformed vectors graphically. The set of given vectors had included eigenvectors, those whose direction would not be changed by being operated upon by one of the matrices. During the wrap-up discussion, the students had considered similarities and differences in how the various matrices had transformed the same given vectors in various ways (SEE <html> <a href="http://www.physics.oregonstate.edu/portfolioswiki/doku.php?id=activities:main&file=lineartrans">Instructor's Guide for Linear Transformations Activity</a> </html> and [[.:lineartranslong|Narrative for Linear Transformations Activity]]). Corinne had designed Day 2’s problems to develop the idea that an eigenvector is an arrow in space whose direction is not changed by a transformation. | + | * Day 2 had been a double period. First the students worked in small groups. Each group used a given matrix to transform a set of given vectors and then plotted the transformed vectors graphically. The set of given vectors had included eigenvectors, those whose direction would not be changed by being operated upon by one of the matrices. During the wrap-up discussion, the students had considered similarities and differences in how the various matrices had transformed the same given vectors in various ways (SEE <html> <a href="http://www.physics.oregonstate.edu/portfolioswiki/doku.php?id=activities:main&file=prlineartrans">Instructor's Guide for Linear Transformations Activity</a> </html> and [[.:lineartranslong|Narrative for Linear Transformations Activity]]). Corinne had designed Day 2’s problems to develop the idea that an eigenvector is an arrow in space whose direction is not changed by a transformation. |

* Day 3 had been a 50-minute interactive lecture that presented properties of rotation matrices in two and three dimensions. | * Day 3 had been a 50-minute interactive lecture that presented properties of rotation matrices in two and three dimensions. | ||

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* Day 5 would complete the Preface with a 50-minute interactive lecture that presented the abstract properties of vector spaces and linear transformations. | * Day 5 would complete the Preface with a 50-minute interactive lecture that presented the abstract properties of vector spaces and linear transformations. | ||

- | The goals of Day 4’s small group activity and wrap-up discussion were for the students to learn an algebraic procedure for finding eigenvalues and eigenvectors for a matrix and to see that the vectors that they were getting from their algebraic calculations were the same ones that they had seen to be unchanged when plotted graphically on Day 2. (See the <html> <a href="http://www.physics.oregonstate.edu/portfolioswiki/doku.php?id=activities:main&file=eigenvectors">Instructor's Guide</a> </html> for the Eigenvectors and Eigenvalues activity.) By comparing and contrasting the solutions presented during the wrap-discussion, the students could gain a deeper understanding of the relations between geometric and algebraic representations of eigenvectors. During their study of quantum mechanics in the weeks ahead, Corinne wanted the students to understand that finding eigenfunctions for the Hamiltonian would mean looking for the functions that are unchanged or just scaled by a constant. | + | The goals of Day 4’s small group activity and wrap-up discussion were for the students to learn an algebraic procedure for finding eigenvalues and eigenvectors for a matrix and to see that the vectors that they were getting from their algebraic calculations were the same ones that they had seen to be unchanged when plotted graphically on Day 2. (See the <html> <a href="http://www.physics.oregonstate.edu/portfolioswiki/doku.php?id=activities:main&file=preigenvectors">Instructor's Guide</a> </html> for the Eigenvectors and Eigenvalues activity.) By comparing and contrasting the solutions presented during the wrap-discussion, the students could gain a deeper understanding of the relations between geometric and algebraic representations of eigenvectors. During their study of quantum mechanics in the weeks ahead, Corinne wanted the students to understand that finding eigenfunctions for the Hamiltonian would mean looking for the functions that are unchanged or just scaled by a constant. |

The six small groups of students had worked on large whiteboards at their tables as shown in Figure 1. | The six small groups of students had worked on large whiteboards at their tables as shown in Figure 1. | ||

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When Corinne called the small groups together for the wrap-up discussion, she was quite explicit in stating her intentions, that she had demonstrated an easy generic case for them and then had engaged the small groups in cases “that have something special about them…that you need to notice.” All of the examples that the students had been working on in their small groups were ones that had either stumped her or her students in the past. Her intent in the wrap-up discussion was to bring out the small number but frequently encountered ways of being stumped in solving for a matrix’s eigenvalues and eigenvectors. | When Corinne called the small groups together for the wrap-up discussion, she was quite explicit in stating her intentions, that she had demonstrated an easy generic case for them and then had engaged the small groups in cases “that have something special about them…that you need to notice.” All of the examples that the students had been working on in their small groups were ones that had either stumped her or her students in the past. Her intent in the wrap-up discussion was to bring out the small number but frequently encountered ways of being stumped in solving for a matrix’s eigenvalues and eigenvectors. | ||

+ | |||

+ | {{whitepapers:narratives:080110day2hour2classroomlayout.png?400 |Figure 2: Classroom Layout During Group Presentations.}} | ||

In acknowledging the challenge of the work the students had been doing, Corinne also affirmed their image of themselves as future physicists, “if you're going to have problems, I wanted you to have them here in class and not two years from now when you're trying to solve one of these by yourself in a job somewhere.” Then she issued a direct order “So I want you to pay attention to the presentations, particularly today, pay attention to the presentations of all of the groups because all of these have some funny little nuance to them.” | In acknowledging the challenge of the work the students had been doing, Corinne also affirmed their image of themselves as future physicists, “if you're going to have problems, I wanted you to have them here in class and not two years from now when you're trying to solve one of these by yourself in a job somewhere.” Then she issued a direct order “So I want you to pay attention to the presentations, particularly today, pay attention to the presentations of all of the groups because all of these have some funny little nuance to them.” | ||

- | As shown in Figure 2 [00:44:19.14], each group’s presenter placed the group’s white board on the ledge of the blackboard in the front of the room so that all could see their work. The other students remained in small groups at their tables. Corinne sat at a table near the back of the room. By placing herself within the class, she ensured that in talking to her, the presenter also would be addressing the entire class. From this position, she also could engage the entire class in conversation as needed. | + | As shown in Figure 2, each group’s presenter placed the group’s white board on the ledge of the blackboard in the front of the room so that all could see their work. The other students remained in small groups at their tables. Corinne sat at a table near the back of the room. By placing herself within the class, she ensured that in talking to her, the presenter also would be addressing the entire class. From this position, she also could engage the entire class in conversation as needed. |

==== Calculating and Representing Complex Eigenvalues and Eigenvectors: Group 1 [00:23:00.17] - [00:30:23.28] ==== | ==== Calculating and Representing Complex Eigenvalues and Eigenvectors: Group 1 [00:23:00.17] - [00:30:23.28] ==== | ||

The first group to present had produced the solution shown in Figure 3 [00:24:27.26] for the two dimensional matrix $A_1$. The group’s presenter began by identifying the matrix they had been given, “So this is, $A$ one is our matrix: zero, minus one, one, zero.” | The first group to present had produced the solution shown in Figure 3 [00:24:27.26] for the two dimensional matrix $A_1$. The group’s presenter began by identifying the matrix they had been given, “So this is, $A$ one is our matrix: zero, minus one, one, zero.” | ||

+ | |||

+ | {{ whitepapers:narratives:080110day2hour2group1.png?400|Figure 3: Whiteboard Presented by Group 1.}} | ||

Corinne interrupted to make a connection to the problems they had worked on earlier in the week, “Before you go on, this is one of the one's from Tuesday's activity; what group had this transformation?” She wanted the students to think back to what they had done in the previous activity in order to bring the resources that they had developed there to this discussion. By making such connections explicit, she was modeling ways of thinking that she wanted the students to adopt, seeking prior knowledge that can be brought to bear on the current focus of attention. | Corinne interrupted to make a connection to the problems they had worked on earlier in the week, “Before you go on, this is one of the one's from Tuesday's activity; what group had this transformation?” She wanted the students to think back to what they had done in the previous activity in order to bring the resources that they had developed there to this discussion. By making such connections explicit, she was modeling ways of thinking that she wanted the students to adopt, seeking prior knowledge that can be brought to bear on the current focus of attention. | ||

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Although Corinne had intended to get students trying to think about complex vector spaces, she does not remember planning to ask this specific question. However, a check of the video of this wrap-up discussion during an earlier year showed a student asking a question about graphing complex eigenvectors. A memory of this question in this context may have cued Corinne’s spontaneous decision here to get students to extend their understanding of real vectors, which can be drawn on the board, to complex vectors, which in general cannot. Corinne considers the conversation that followed an example of a particularly productive classroom conversation because different students had different ways of coming to terms with the complication that was posed and as the students expressed their various different ways of thinking about it other students in the class heard enough alternative explanations that hopefully at least one resonated with them. | Although Corinne had intended to get students trying to think about complex vector spaces, she does not remember planning to ask this specific question. However, a check of the video of this wrap-up discussion during an earlier year showed a student asking a question about graphing complex eigenvectors. A memory of this question in this context may have cued Corinne’s spontaneous decision here to get students to extend their understanding of real vectors, which can be drawn on the board, to complex vectors, which in general cannot. Corinne considers the conversation that followed an example of a particularly productive classroom conversation because different students had different ways of coming to terms with the complication that was posed and as the students expressed their various different ways of thinking about it other students in the class heard enough alternative explanations that hopefully at least one resonated with them. | ||

- | In response to Corinne’s request, the first group’s presenter drew two perpendicular axes on the blackboard and labeled them $x$ (horizontal) and $i$ (vertical). Next he drew an arrow downward and to the left into the third quadrant. Another student proposed a correction that caused the presenter to erase the arrow and redraw it upward and to the left into the second quadrant. After further coaching from a classmate, he erased the second arrow and drew a third version downward and to the right into the fourth quadrant. | + | In response to Corinne’s request, the first group’s presenter drew two perpendicular axes on the blackboard and labeled them $x$ (horizontal) and $i$ (vertical). Next he drew an arrow downward and to the left into the third quadrant (see Figure 4). |

+ | {{whitepapers:narratives:080110day2hour2group1drawing.png?400 |Figure 4: A classmate proposes a correction to the Group 1 presenter's eigenvector drawing.}} | ||

+ | Another student proposed a correction that caused the presenter to erase the arrow and redraw it upward and to the left into the second quadrant. After further coaching from a classmate, he erased the second arrow and drew a third version downward and to the right into the fourth quadrant. | ||

After the first group’s presenter turned and looked at Corinne for confirmation, she deflected the authority of judging right answers to the students by asking “Do you all agree with that?” A student responded, “You need four dimensions for more-than-one-component vectors.” Apparently she had remembered and understood a related discussion two days earlier. Corinne elaborated this statement, “You need four dimensions. [Student J] says that you need four dimensions and you've only got two” and then queried, “So why do you need four dimensions?” Several students contributed their ideas in overlapping talk. Corinne affirmed these with, “Absolutely! This is a two dimensional piece of a four dimensional space.” | After the first group’s presenter turned and looked at Corinne for confirmation, she deflected the authority of judging right answers to the students by asking “Do you all agree with that?” A student responded, “You need four dimensions for more-than-one-component vectors.” Apparently she had remembered and understood a related discussion two days earlier. Corinne elaborated this statement, “You need four dimensions. [Student J] says that you need four dimensions and you've only got two” and then queried, “So why do you need four dimensions?” Several students contributed their ideas in overlapping talk. Corinne affirmed these with, “Absolutely! This is a two dimensional piece of a four dimensional space.” | ||

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==== Representing a Generalized Sense of Direction: Group 2 [00:30:23.28] – [00:33:37.19] ==== | ==== Representing a Generalized Sense of Direction: Group 2 [00:30:23.28] – [00:33:37.19] ==== | ||

- | The presenter for the second group began by pointing to the matrix at the top left of his group’s white board, “so we had this one right here: zero, one, one, zero.” This group’s matrix and solution are shown in Figure 4. [00:30:41.15] | + | The presenter for the second group began by pointing to the matrix at the top left of his group’s white board, “so we had this one right here: zero, one, one, zero.” This group’s matrix and solution are shown in Figure 5. |

+ | | ||

+ | {{ whitepapers:narratives:080110hour2group2.png?400|Figure 5: Whiteboard Presented by Group 2.}} | ||

Corinne again interrupted to get the class to anticipate what the answer was going to be by asking “Who had this one on Tuesday?” After a long pause amid low student talk and laughter, she relented, “Nobody's going to admit to it? Okay. So everybody then, what, what does this matrix do to real vectors?” A student responded, “It reflects” and Corinne pressed for more specifics, “It reflects along what line?” Students offered, $y=x$, $y$ axis” Corinne responded, “It reflects along $y =x$” and asked the next appropriate question for reviewing this kind of problem, “So what do we expect to be eigenvectors?” and a student answered, “$y=x$; one, one.” | Corinne again interrupted to get the class to anticipate what the answer was going to be by asking “Who had this one on Tuesday?” After a long pause amid low student talk and laughter, she relented, “Nobody's going to admit to it? Okay. So everybody then, what, what does this matrix do to real vectors?” A student responded, “It reflects” and Corinne pressed for more specifics, “It reflects along what line?” Students offered, $y=x$, $y$ axis” Corinne responded, “It reflects along $y =x$” and asked the next appropriate question for reviewing this kind of problem, “So what do we expect to be eigenvectors?” and a student answered, “$y=x$; one, one.” | ||

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Corinne responded by requesting an elaboration, “Okay. So you have one that is along $x$ $y$, along the line $x = y$, but what's that other one?” When the presenter responded, “Ah that is the one that is 90 degrees perpendicular to,” Corinne said softly, “pi over two” and the student corrected himself, “pi over two, same thing’ with a laugh and continued, “Ah, it's pi over two shifted.” However, he encountered some uncertainly “cause eigenvalues can change” and thinking on his feet concluded, “well, it's just a rev - it's a negative of itself, it's ninety, ah, pi shifted from itself, which makes it just the opposite.” He seemed to be choosing his language very carefully to avoid saying that this was an eigenvector because he did not seem to be sure what “changing direction” meant. In reflecting upon the video of this interaction, Corinne noted that the issue is that common everyday language says that the negative of a vector points in the opposite direction but in technical language one would say that the negative of a vector points in the ‘same’ direction but it’s length is multiplied by minus one. This example provided a good opportunity to have that discussion with the students. | Corinne responded by requesting an elaboration, “Okay. So you have one that is along $x$ $y$, along the line $x = y$, but what's that other one?” When the presenter responded, “Ah that is the one that is 90 degrees perpendicular to,” Corinne said softly, “pi over two” and the student corrected himself, “pi over two, same thing’ with a laugh and continued, “Ah, it's pi over two shifted.” However, he encountered some uncertainly “cause eigenvalues can change” and thinking on his feet concluded, “well, it's just a rev - it's a negative of itself, it's ninety, ah, pi shifted from itself, which makes it just the opposite.” He seemed to be choosing his language very carefully to avoid saying that this was an eigenvector because he did not seem to be sure what “changing direction” meant. In reflecting upon the video of this interaction, Corinne noted that the issue is that common everyday language says that the negative of a vector points in the opposite direction but in technical language one would say that the negative of a vector points in the ‘same’ direction but it’s length is multiplied by minus one. This example provided a good opportunity to have that discussion with the students. | ||

- | Again Corinne requested a graphical representation, “so can you draw those?” As shown in Figure 5 [00:33:12:06], the presenter for group two drew two axes and a dotted line representing y = x, drew an arrow from the origin along $y=x$ into the first quadrant for the first eigenvector, [$v_1 =\pmatrix{1\cr 1}$ ] and then drew an arrow from the origin to the fourth quadrant, for the second eigenvector [$v_2 =\pmatrix{1\cr -1}$]. Corinne offered an interpretation, “So one, one goes to itself and one, negative one goes to minus itself.” [$A\vert v_1\rangle = 1\vert v_1\rangle$] and [$A\vert v_2\rangle = -1\vert v_2\rangle$]. The presenter for group two turned toward Corinne and nodded. Corinne continued, “Good. So this is why we want to include multiplication by negative one as still being, pointing in the same direction in quotes [makes signs for quotation marks with both arms raised] our generalized sense of direction, because it is a solution to the eigenvalue equation.” | + | {{whitepapers:narratives:080110day2hour2group2drawing.png?400 |Figure 6: Presenter from Group 2 draws his group's eigenvectors.}} |

+ | | ||

+ | Again Corinne requested a graphical representation, “so can you draw those?” As shown in Figure 6, the presenter for group two drew two axes and a dotted line representing y = x, drew an arrow from the origin along $y=x$ into the first quadrant for the first eigenvector, [$v_1 =\pmatrix{1\cr 1}$ ] and then drew an arrow from the origin to the fourth quadrant, for the second eigenvector [$v_2 =\pmatrix{1\cr -1}$]. Corinne offered an interpretation, “So one, one goes to itself and one, negative one goes to minus itself.” [$A\vert v_1\rangle = 1\vert v_1\rangle$] and [$A\vert v_2\rangle = -1\vert v_2\rangle$]. The presenter for group two turned toward Corinne and nodded. Corinne continued, “Good. So this is why we want to include multiplication by negative one as still being, pointing in the same direction in quotes [makes signs for quotation marks with both arms raised] our generalized sense of direction, because it is a solution to the eigenvalue equation.” | ||

==== Monitoring Algebraic Accuracy and Interpreting a Result: Group 3 [00:33:37.19]- [00:37:15.21] ==== | ==== Monitoring Algebraic Accuracy and Interpreting a Result: Group 3 [00:33:37.19]- [00:37:15.21] ==== | ||

- | The presenter for the third group began by pointing to the matrix they had used and saying, “So the thing in purple is $A$ three. Our matrix is negative one, zero, zero, negative one.” Group 3’s matrix and solution are shown in Figure 6. [00:33:51:15] | + | The presenter for the third group began by pointing to the matrix they had used and saying, “So the thing in purple is $A$ three. Our matrix is negative one, zero, zero, negative one.” Group 3’s matrix and solution are shown in Figure 7. |

+ | | ||

+ | {{ whitepapers:narratives:080110hour2group3.png?400|Figure 7: Whiteboard Presented by Group 3.}} | ||

The presenter rapidly described her group’s calculations and presented their result, “our lambda value is one” and holding up one finger for emphasis, she reiterated, “we only have one.” Modeling what a professional would ask if anticipating two solutions, Corinne questioned that result, “How can that be if it was a quadratic equation?” | The presenter rapidly described her group’s calculations and presented their result, “our lambda value is one” and holding up one finger for emphasis, she reiterated, “we only have one.” Modeling what a professional would ask if anticipating two solutions, Corinne questioned that result, “How can that be if it was a quadratic equation?” | ||

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==== Noticing and Defining Degeneracy: Group Four [00:37:15.21] - [00:43:17.28] ==== | ==== Noticing and Defining Degeneracy: Group Four [00:37:15.21] - [00:43:17.28] ==== | ||

- | As Corinne returned to her seat near the back of the room, she made a decision to call on a particular group, one that she wanted to follow the previous example, so that the class could continue the conversation they were having. The presenter for group four began in the same way as the others, by identifying the matrix his group had used, “Our matrix was negative one, zero, zero, zero, negative one - this thing here” he finally said as he pointed to a three by three matrix with zeros everywhere but on the diagonal. Group 4’s matrix and solution are shown in Figure 7. [00:39:29.13] | + | As Corinne returned to her seat near the back of the room, she made a decision to call on a particular group, one that she wanted to follow the previous example, so that the class could continue the conversation they were having. The presenter for group four began in the same way as the others, by identifying the matrix his group had used, “Our matrix was negative one, zero, zero, zero, negative one - this thing here” he finally said as he pointed to a three by three matrix with zeros everywhere but on the diagonal. Group 4’s matrix and solution are shown in Figure 8. |

+ | | ||

+ | {{whitepapers:narratives:080110hour2group4.png?400 |Figure 8: Whiteboard Presented by Group 4.}} | ||

Corinne took this opportunity to coach his use of language, “Lots of times people, if they have a matrix that has things only on the diagonal, instead of saying minus one, zero, zero, zero, minus one, you got tired right? and quit in the middle? <yeah> <laughter> which is what everybody does. If it's just diagonal, the convention is to say it's diagonal and you just read off the diagonal ones, and then everybody knows its diagonal.” | Corinne took this opportunity to coach his use of language, “Lots of times people, if they have a matrix that has things only on the diagonal, instead of saying minus one, zero, zero, zero, minus one, you got tired right? and quit in the middle? <yeah> <laughter> which is what everybody does. If it's just diagonal, the convention is to say it's diagonal and you just read off the diagonal ones, and then everybody knows its diagonal.” | ||

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Acknowledging that class was almost over, Corinne asked for the group “with $a$’s and $d$’s” to “give us a thirty second presentation” and returned to her chair in the back of the room. | Acknowledging that class was almost over, Corinne asked for the group “with $a$’s and $d$’s” to “give us a thirty second presentation” and returned to her chair in the back of the room. | ||

- | The presenter for this group began by using the language about diagonal matrices that Corinne just had coached during the presentation by group 4, “So we had a diagonal matrix with $a$ and $d$, so we went through and we used the general equation to solve for the eigenvalues which were $a$ and $d$ cause you know all these reasons anyways” as he waved his hands over various equations. Group 5’s matrix and solution are shown in Figure 8 [00:43:49.00]. | + | The presenter for this group began by using the language about diagonal matrices that Corinne just had coached during the presentation by group 4, “So we had a diagonal matrix with $a$ and $d$, so we went through and we used the general equation to solve for the eigenvalues which were $a$ and $d$ cause you know all these reasons anyways” as he waved his hands over various equations. Group 5’s matrix and solution are shown in Figure 9. |

+ | | ||

+ | {{ whitepapers:narratives:080110hour2group5.png?400|Figure 9: Whiteboard Presented by Group 5.}} | ||

In spite of the limited time available, Corinne intervened, “Wait! Are those reasons obvious?” The student responded, “Well, solving the quadratic [ $(\lambda - a)(\lambda -d) = 0$], Liz pointed out you don't really need to solve it when you know that $a$ minus $a$ is zero and $d$ minus $d$ is zero.” | In spite of the limited time available, Corinne intervened, “Wait! Are those reasons obvious?” The student responded, “Well, solving the quadratic [ $(\lambda - a)(\lambda -d) = 0$], Liz pointed out you don't really need to solve it when you know that $a$ minus $a$ is zero and $d$ minus $d$ is zero.” | ||

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van Zee, E. H. & Minstrell, J. (1997b). Using questioning to guide student thinking. The Journal of the Learning Sciences, 6, 229-271. | van Zee, E. H. & Minstrell, J. (1997b). Using questioning to guide student thinking. The Journal of the Learning Sciences, 6, 229-271. | ||

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- | Figures (and possible clips) | ||

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- | Figure 1: Small group working on white board [00:11:28.21 | ||

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- | Figure 2: Presenter presenting small group’s work on white board to instructor and class [00:44:19.14] | ||

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- | Figure 3: Group 1’s matrix and solution [00:24:27.26] | ||

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- | Figure 4: Group 2’s matrix and solution [00:30:41.15] | ||

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- | Figure 5: Graphical representation of Group 2’s eigenvectors [00:33:12:06] | ||

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- | Figure 6: Group 3’s matrix and solution [00:33:51:15] | ||

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- | Figure 7: Group 4’s matrix and solution [00:39:29.13] | ||

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- | Figure 8: Group 5’s matrix and solution [00:43:49.00] | ||