Finding an Idea for Exercise: Let's Explore Vector Spaces!

In summary: I think you need to talk to your teacher about this.In summary, the person is looking for help with defining a family of vectors that satisfies specific criteria in relation to a given vector space and basis. The criteria and purpose of this family of vectors are unclear and may require further clarification.
  • #1
math6
67
0
Hello friends, I am looking for an idea to my exercise!

let's E be a vector space, e_ {i} be a basis of E, b_ {a} an element of E then

b_ {a} = b_ {a} ^ {i} e_ {i}.

I want to define a family of vectors {t_ {i}}, that lives on E , (how to choose this family already, it must not be a base or even generator?!) And I want to write b_ {a} in function of {t_ {i}}?!

I hope you can help me.
Good day :)...
 
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  • #2
I don't understand. Is this a textbook problem? If so, can you post the full statement of the problem? If not, can you provide some context?

From what you have told us, it seems to me that you want to find a set of vectors ##\{t_i\}## with the property that for each vector b (I dropped the subscript a, because I don't see why it's there), there's a function ##f_b## such that ##f_b(t_1,\dots,t_n)=b##. And for some unspecified reason, it's important that the left-hand side of that last equality isn't a linear combination of the ##t_i##.

There must be something more to this problem, because why not just let ##f_b## be the (constant) function such that ##f_b(x_1,\dots,x_n)=b## for all ##x_1,\dots,x_n## in the vector space? Then any choice of the ##t_1,\dots,t_n## will do.
 
  • #3
I have moved this to the "Linear and Abstract Algebra" forum.

What, exactly, do you want to do. You say you are given a basis for a vector space and "want to define a family of vectors".

Is this family supposed to have any special property? There are an infinite number of ways to write a family of vectors in terms of a basis. A "one parameter" family would be of the for [itex]\{f_1(t) e_1+ f_2(t)e_2+ \cdot\cdot\cdot+ f_n(t)e_n\}[/itex] where n is the dimension of the space and [itex]f_i(t)[/itex] are n specific functions of the parameter t. We could similarly define "two parameter", etc. families of vectors. Your question is just too general to have a simple answer.
 

1. What is a vector space?

A vector space is a mathematical concept that refers to a set of objects (vectors) that can be added together and multiplied by a scalar to obtain another vector in the same set. It is a fundamental concept in linear algebra and is used in many fields of science, including physics, engineering, and computer science.

2. How can vector spaces be applied to exercise?

Vector spaces can be applied to exercise by using them as a framework for understanding and creating exercise routines. Different types of exercises can be represented as vectors, and their combinations can be thought of as vector additions. Additionally, scalar multiplication can be used to adjust the intensity or difficulty of an exercise.

3. How can one come up with new ideas for exercise using vector spaces?

One way to come up with new ideas for exercise using vector spaces is by exploring different combinations of exercises and thinking about how they can be represented as vectors. Another approach is to use vector operations, such as addition, subtraction, and multiplication, to modify existing exercises and create new variations.

4. What are the benefits of using vector spaces in exercise?

Using vector spaces in exercise can help individuals structure their workouts and create a balanced routine that targets different muscle groups. It also allows for creativity and variety in exercise routines, preventing boredom and plateaus. Additionally, understanding vector spaces can improve one's understanding of the mechanics and effectiveness of different exercises.

5. Are there any drawbacks to using vector spaces in exercise?

One potential drawback of using vector spaces in exercise is that it may be more challenging for beginners to understand and apply. Additionally, it may not be suitable for certain types of exercises, such as cardio or flexibility training. However, with practice and understanding, vector spaces can be a valuable tool for creating personalized and effective exercise routines.

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