Eigenvalues and Eigenvectors
About points...
We associate a certain number of points with each exercise.
When you click an exercise into a collection, this number will be taken as points for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit the number of points for the exercise in the collection independently, without any effect on "points by default" as represented by the number here.
That being said... How many "default points" should you associate with an exercise upon creation?
As with difficulty, there is no straight forward and generally accepted way.
But as a guideline, we tend to give as many points by default as there are mathematical steps to do in the exercise.
Again, very vague... But the number should kind of represent the "work" required.
When you click an exercise into a collection, this number will be taken as points for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit the number of points for the exercise in the collection independently, without any effect on "points by default" as represented by the number here.
That being said... How many "default points" should you associate with an exercise upon creation?
As with difficulty, there is no straight forward and generally accepted way.
But as a guideline, we tend to give as many points by default as there are mathematical steps to do in the exercise.
Again, very vague... But the number should kind of represent the "work" required.
About difficulty...
We associate a certain difficulty with each exercise.
When you click an exercise into a collection, this number will be taken as difficulty for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit its difficulty in the collection independently, without any effect on the "difficulty by default" here.
Why we use chess pieces? Well... we like chess, we like playing around with \(\LaTeX\)-fonts, we wanted symbols that need less space than six stars in a table-column... But in your layouts, you are of course free to indicate the difficulty of the exercise the way you want.
That being said... How "difficult" is an exercise? It depends on many factors, like what was being taught etc.
In physics exercises, we try to follow this pattern:
Level 1 - One formula (one you would find in a reference book) is enough to solve the exercise. Example exercise
Level 2 - Two formulas are needed, it's possible to compute an "in-between" solution, i.e. no algebraic equation needed. Example exercise
Level 3 - "Chain-computations" like on level 2, but 3+ calculations. Still, no equations, i.e. you are not forced to solve it in an algebraic manner. Example exercise
Level 4 - Exercise needs to be solved by algebraic equations, not possible to calculate numerical "in-between" results. Example exercise
Level 5 -
Level 6 -
When you click an exercise into a collection, this number will be taken as difficulty for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit its difficulty in the collection independently, without any effect on the "difficulty by default" here.
Why we use chess pieces? Well... we like chess, we like playing around with \(\LaTeX\)-fonts, we wanted symbols that need less space than six stars in a table-column... But in your layouts, you are of course free to indicate the difficulty of the exercise the way you want.
That being said... How "difficult" is an exercise? It depends on many factors, like what was being taught etc.
In physics exercises, we try to follow this pattern:
Level 1 - One formula (one you would find in a reference book) is enough to solve the exercise. Example exercise
Level 2 - Two formulas are needed, it's possible to compute an "in-between" solution, i.e. no algebraic equation needed. Example exercise
Level 3 - "Chain-computations" like on level 2, but 3+ calculations. Still, no equations, i.e. you are not forced to solve it in an algebraic manner. Example exercise
Level 4 - Exercise needs to be solved by algebraic equations, not possible to calculate numerical "in-between" results. Example exercise
Level 5 -
Level 6 -
Question
Solution
Short
Video
\(\LaTeX\)
No explanation / solution video to this exercise has yet been created.
Visit our YouTube-Channel to see solutions to other exercises.
Don't forget to subscribe to our channel, like the videos and leave comments!
Visit our YouTube-Channel to see solutions to other exercises.
Don't forget to subscribe to our channel, like the videos and leave comments!
Exercise:
A system of linear differential s is given by the matrix bf A pmatrix - & & pmatrix abcliste abc Determine the eigenvalues. abc Verify that the eigenvectors are vec v_ pmatrix pmatrix quad textrmand quad vec v_ pmatrix pmatrix abc Discuss the stability of the system and sketch the phase portrait. abcliste
Solution:
abcliste abc The trace and determinant are tau -+ - Delta - - - and thus the eigenvalues are lambda_ fractaupmsqrttau^-Delta frac-pmsqrt+ frac-pm Longrightarrow lambda_ lambda_ - abc bf Avec v_ pmatrix - & & pmatrix pmatrix pmatrix pmatrix - + + pmatrix pmatrix pmatrix vec v_ bf Avec v_ pmatrix - & & pmatrix pmatrix pmatrix pmatrix - + + pmatrix pmatrix - pmatrix -vec v_ This proves that vec v_ is the eigenvector for eigenvalue lambda_ and vec v_ the eigenvector for eigenvalue lambda_-. abc Since both eigenvalues are real with lambda_ and lambda_ the fixed po at the origin is a saddle po. The eigenvector vec v_ defines the straight-line solution with exponential growth moving away from the origin whereas vec v_ defines the straight-line solution with exponential decrease moving towards the origin. vspacemm The orbites cannot cross the two straight-lines defined by the eigenvectors. Near the horizontal axis they are directed towards the origin but then turn away and approach the tilted line. center includegraphicswidthtextwidth#image_path:phasportrait-# center abcliste
A system of linear differential s is given by the matrix bf A pmatrix - & & pmatrix abcliste abc Determine the eigenvalues. abc Verify that the eigenvectors are vec v_ pmatrix pmatrix quad textrmand quad vec v_ pmatrix pmatrix abc Discuss the stability of the system and sketch the phase portrait. abcliste
Solution:
abcliste abc The trace and determinant are tau -+ - Delta - - - and thus the eigenvalues are lambda_ fractaupmsqrttau^-Delta frac-pmsqrt+ frac-pm Longrightarrow lambda_ lambda_ - abc bf Avec v_ pmatrix - & & pmatrix pmatrix pmatrix pmatrix - + + pmatrix pmatrix pmatrix vec v_ bf Avec v_ pmatrix - & & pmatrix pmatrix pmatrix pmatrix - + + pmatrix pmatrix - pmatrix -vec v_ This proves that vec v_ is the eigenvector for eigenvalue lambda_ and vec v_ the eigenvector for eigenvalue lambda_-. abc Since both eigenvalues are real with lambda_ and lambda_ the fixed po at the origin is a saddle po. The eigenvector vec v_ defines the straight-line solution with exponential growth moving away from the origin whereas vec v_ defines the straight-line solution with exponential decrease moving towards the origin. vspacemm The orbites cannot cross the two straight-lines defined by the eigenvectors. Near the horizontal axis they are directed towards the origin but then turn away and approach the tilted line. center includegraphicswidthtextwidth#image_path:phasportrait-# center abcliste
Meta Information
Exercise:
A system of linear differential s is given by the matrix bf A pmatrix - & & pmatrix abcliste abc Determine the eigenvalues. abc Verify that the eigenvectors are vec v_ pmatrix pmatrix quad textrmand quad vec v_ pmatrix pmatrix abc Discuss the stability of the system and sketch the phase portrait. abcliste
Solution:
abcliste abc The trace and determinant are tau -+ - Delta - - - and thus the eigenvalues are lambda_ fractaupmsqrttau^-Delta frac-pmsqrt+ frac-pm Longrightarrow lambda_ lambda_ - abc bf Avec v_ pmatrix - & & pmatrix pmatrix pmatrix pmatrix - + + pmatrix pmatrix pmatrix vec v_ bf Avec v_ pmatrix - & & pmatrix pmatrix pmatrix pmatrix - + + pmatrix pmatrix - pmatrix -vec v_ This proves that vec v_ is the eigenvector for eigenvalue lambda_ and vec v_ the eigenvector for eigenvalue lambda_-. abc Since both eigenvalues are real with lambda_ and lambda_ the fixed po at the origin is a saddle po. The eigenvector vec v_ defines the straight-line solution with exponential growth moving away from the origin whereas vec v_ defines the straight-line solution with exponential decrease moving towards the origin. vspacemm The orbites cannot cross the two straight-lines defined by the eigenvectors. Near the horizontal axis they are directed towards the origin but then turn away and approach the tilted line. center includegraphicswidthtextwidth#image_path:phasportrait-# center abcliste
A system of linear differential s is given by the matrix bf A pmatrix - & & pmatrix abcliste abc Determine the eigenvalues. abc Verify that the eigenvectors are vec v_ pmatrix pmatrix quad textrmand quad vec v_ pmatrix pmatrix abc Discuss the stability of the system and sketch the phase portrait. abcliste
Solution:
abcliste abc The trace and determinant are tau -+ - Delta - - - and thus the eigenvalues are lambda_ fractaupmsqrttau^-Delta frac-pmsqrt+ frac-pm Longrightarrow lambda_ lambda_ - abc bf Avec v_ pmatrix - & & pmatrix pmatrix pmatrix pmatrix - + + pmatrix pmatrix pmatrix vec v_ bf Avec v_ pmatrix - & & pmatrix pmatrix pmatrix pmatrix - + + pmatrix pmatrix - pmatrix -vec v_ This proves that vec v_ is the eigenvector for eigenvalue lambda_ and vec v_ the eigenvector for eigenvalue lambda_-. abc Since both eigenvalues are real with lambda_ and lambda_ the fixed po at the origin is a saddle po. The eigenvector vec v_ defines the straight-line solution with exponential growth moving away from the origin whereas vec v_ defines the straight-line solution with exponential decrease moving towards the origin. vspacemm The orbites cannot cross the two straight-lines defined by the eigenvectors. Near the horizontal axis they are directed towards the origin but then turn away and approach the tilted line. center includegraphicswidthtextwidth#image_path:phasportrait-# center abcliste
Contained in these collections
| Title | Matched on |
|---|---|
| Eigenvalues and Eigenvectors | title |
| Phase Portrait | tags |
| Phase Portrait | tags |
| Eigenvalues | tags |
| Initial Value Problem | tags |
Similar exercises (8)
| Title | Matched on |
|---|---|
| Eigenvalues and Eigenvectors | title |
| Phase Portrait | tags |
| Phase Portrait | tags |
| Eigenvalues | tags |
| Initial Value Problem | tags |
| Strange Spring | tags |
| Overdamped Oscillation | tags |
| Rabbits & Foxes | tags |