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

  • Thread starter willy0625
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Any point in 3-dimensional space

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

[tex]\sum_{j}c_{j}{\bf v_{j}}[/tex]

can be made arbitrarily small?




My approach was letting

[tex] \parallel{\bf x}\parallel \leq \bigg{\parallel} \sum_{j}c_{j}{\bf v_{j}}\bigg\parallel (*)[/tex]

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

[tex] \parallel{\bf x} - \sum_{j}c_{j}{\bf v_{j}}\bigg\parallel [/tex]

[tex] \leq 4n\sum_{j}|c_{j}|^{2} [/tex]


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

Therefore by letting [tex] \sum_{j}|c_{j}|^{2} \rightarrow 0[/tex],

we get

[tex]\parallel{\bf x}- \sum_{j}c_{j}{\bf v_{j}}\bigg\parallel \leq 0 [/tex]

Thus deducing

[tex] {\bf x} = \sum_{j}c_{j}{\bf v_{j}} [/tex]

I am not sure if assuming (*) is the correct way of doing this since we miss out on the other possibility (>).
 
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LCKurtz
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Is is possible to write any point [tex] {\bf x}\in \mathbb{R}^{3}[/tex] as a linear combination of vectors [tex]{\bf v_{j}}[/tex] inside the unit ball over [tex]\mathbb{R}[/tex] given that

[tex]\sum_{j}c_{j}{\bf v_{j}}[/tex]

can be made arbitrarily small?
I'm guessing you haven't told us everything. Let

[tex]v_1 = \frac x {2||x||} \hbox{ so } ||v_1|| = \frac 1 2[/tex]

Let [itex]c = 2||x||[/itex]. Then and [itex]x = cv_1[/itex].
 

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