Any point in [\tex]mathbb{R}^{3}[\tex]

[willy0625]

Any point in 3-dimensional space

Is is possible to write any point {\bf x}\in \mathbb{R}^{3} as a linear combination of vectors {\bf v_{j}} inside the unit ball over \mathbb{R} given that

\sum_{j}c_{j}{\bf v_{j}}

can be made arbitrarily small?




My approach was letting

\parallel{\bf x}\parallel \leq \bigg{\parallel} \sum_{j}c_{j}{\bf v_{j}}\bigg\parallel (*)

Then applying the triangle inequality to the two terms then (*), the triangle inequality again to the linear combination, \parallel v_{j}\parallel<1 for all n then the Cauchy's inequality to obtain

\parallel{\bf x} - \sum_{j}c_{j}{\bf v_{j}}\bigg\parallel

\leq 4n\sum_{j}|c_{j}|^{2}


where n is the number of terms in the linear combination, which is finite.

Therefore by letting \sum_{j}|c_{j}|^{2} \rightarrow 0,

we get

\parallel{\bf x}- \sum_{j}c_{j}{\bf v_{j}}\bigg\parallel \leq 0

Thus deducing

{\bf x} = \sum_{j}c_{j}{\bf v_{j}}

I am not sure if assuming (*) is the correct way of doing this since we miss out on the other possibility (>).

 

[LCKurtz]

Is is possible to write any point {\bf x}\in \mathbb{R}^{3} as a linear combination of vectors {\bf v_{j}} inside the unit ball over \mathbb{R} given that

\sum_{j}c_{j}{\bf v_{j}}

can be made arbitrarily small?

I'm guessing you haven't told us everything. Let

v_1 = \frac x {2||x||} \hbox{ so } ||v_1|| = \frac 1 2

Let c = 2||x||. Then and x = cv_1.