Cauchy-Schwarz-Ungleichung
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
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Don't forget to subscribe to our channel, like the videos and leave comments!
Exercise:
Sei d in mathbbN und VmathbbC^d. Dann gilt für alle bf v w in V die Ungleichung |langle vwrangle| leq ||v||||w|| auch geschrieben als _i^n |v_i||w_i| leq sqrt _i^n |v_i|^ sqrt _i^n |w_i|^ Des Weiteren gilt Gleichheit genau dann wenn bf v w linear abhängig sind.
Solution:
Für v oder w ist die Cauchy-Schwarz-Ungleichung erfüllt weil leq . Also betrachtet man vw neq . Zu Beginn bemerkt man dass für alle xy in mathbbR gilt x-y^ geq quad Rightarrow xy leq x^+y^ Nun benutzt man die zusammen mit: x:frac|v_i|||v||_ quad x:frac|w_i|||w||_ Man erhält damit xy leq x^+y^ Rightarrow frac|v_i||w_i|||v||_||w||_ leq frac|v_i|^||v||_^+frac|w_i|^||w||_^ Man miert nun auf beiden Seiten über i...n und erhält _i^n frac|v_i||w_i|||v||_||w||_ frac||v||_||w||__i^n |v_i||w_i| leq _i^nfrac|v_i|^||v||_^+_i^nfrac|w_i|^||w||_^ frac||v_i||_^||v||_^+frac||w||_^||w||_^ + Also frac||v||_||w||__i^n |v_i||w_i| leq Rightarrow _i^n |v_i||w_i| leq ||v||||w||
Sei d in mathbbN und VmathbbC^d. Dann gilt für alle bf v w in V die Ungleichung |langle vwrangle| leq ||v||||w|| auch geschrieben als _i^n |v_i||w_i| leq sqrt _i^n |v_i|^ sqrt _i^n |w_i|^ Des Weiteren gilt Gleichheit genau dann wenn bf v w linear abhängig sind.
Solution:
Für v oder w ist die Cauchy-Schwarz-Ungleichung erfüllt weil leq . Also betrachtet man vw neq . Zu Beginn bemerkt man dass für alle xy in mathbbR gilt x-y^ geq quad Rightarrow xy leq x^+y^ Nun benutzt man die zusammen mit: x:frac|v_i|||v||_ quad x:frac|w_i|||w||_ Man erhält damit xy leq x^+y^ Rightarrow frac|v_i||w_i|||v||_||w||_ leq frac|v_i|^||v||_^+frac|w_i|^||w||_^ Man miert nun auf beiden Seiten über i...n und erhält _i^n frac|v_i||w_i|||v||_||w||_ frac||v||_||w||__i^n |v_i||w_i| leq _i^nfrac|v_i|^||v||_^+_i^nfrac|w_i|^||w||_^ frac||v_i||_^||v||_^+frac||w||_^||w||_^ + Also frac||v||_||w||__i^n |v_i||w_i| leq Rightarrow _i^n |v_i||w_i| leq ||v||||w||
Meta Information
Exercise:
Sei d in mathbbN und VmathbbC^d. Dann gilt für alle bf v w in V die Ungleichung |langle vwrangle| leq ||v||||w|| auch geschrieben als _i^n |v_i||w_i| leq sqrt _i^n |v_i|^ sqrt _i^n |w_i|^ Des Weiteren gilt Gleichheit genau dann wenn bf v w linear abhängig sind.
Solution:
Für v oder w ist die Cauchy-Schwarz-Ungleichung erfüllt weil leq . Also betrachtet man vw neq . Zu Beginn bemerkt man dass für alle xy in mathbbR gilt x-y^ geq quad Rightarrow xy leq x^+y^ Nun benutzt man die zusammen mit: x:frac|v_i|||v||_ quad x:frac|w_i|||w||_ Man erhält damit xy leq x^+y^ Rightarrow frac|v_i||w_i|||v||_||w||_ leq frac|v_i|^||v||_^+frac|w_i|^||w||_^ Man miert nun auf beiden Seiten über i...n und erhält _i^n frac|v_i||w_i|||v||_||w||_ frac||v||_||w||__i^n |v_i||w_i| leq _i^nfrac|v_i|^||v||_^+_i^nfrac|w_i|^||w||_^ frac||v_i||_^||v||_^+frac||w||_^||w||_^ + Also frac||v||_||w||__i^n |v_i||w_i| leq Rightarrow _i^n |v_i||w_i| leq ||v||||w||
Sei d in mathbbN und VmathbbC^d. Dann gilt für alle bf v w in V die Ungleichung |langle vwrangle| leq ||v||||w|| auch geschrieben als _i^n |v_i||w_i| leq sqrt _i^n |v_i|^ sqrt _i^n |w_i|^ Des Weiteren gilt Gleichheit genau dann wenn bf v w linear abhängig sind.
Solution:
Für v oder w ist die Cauchy-Schwarz-Ungleichung erfüllt weil leq . Also betrachtet man vw neq . Zu Beginn bemerkt man dass für alle xy in mathbbR gilt x-y^ geq quad Rightarrow xy leq x^+y^ Nun benutzt man die zusammen mit: x:frac|v_i|||v||_ quad x:frac|w_i|||w||_ Man erhält damit xy leq x^+y^ Rightarrow frac|v_i||w_i|||v||_||w||_ leq frac|v_i|^||v||_^+frac|w_i|^||w||_^ Man miert nun auf beiden Seiten über i...n und erhält _i^n frac|v_i||w_i|||v||_||w||_ frac||v||_||w||__i^n |v_i||w_i| leq _i^nfrac|v_i|^||v||_^+_i^nfrac|w_i|^||w||_^ frac||v_i||_^||v||_^+frac||w||_^||w||_^ + Also frac||v||_||w||__i^n |v_i||w_i| leq Rightarrow _i^n |v_i||w_i| leq ||v||||w||
Contained in these collections
| Title | Matched on |
|---|---|
| Euklidische Norm | tags |
| Teleskopprodukt | tags |
| Binomischer Lehrsatz | tags |
| Kardinalität der Menge der Permutationen einer endlichen Menge | tags |
| Teleskopsumme | tags |