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

[math6]

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 :)...

 

[Fredrik]

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.

 

[HallsofIvy]

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 \{f_1(t) e_1+ f_2(t)e_2+ \cdot\cdot\cdot+ f_n(t)e_n\} where n is the dimension of the space and f_i(t) 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.