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 $x = cv_1$.