(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

If we have a normed vector space, and a sequence of vectors

[itex]\{\mathbf{v}_k\}_{k=1}^{N}[/itex] in the normed vector space.

If there exists a constant B>0 such that the following holds for all scalar coefficients [itex]c_1,c_2\cdots c_N[/itex]

[itex]B\sum\limits_{k=1}^N |c_k|^2 \leq ||\sum\limits_{k=1}^Nc_k\mathbf{v}_k||^2[/itex]

Show that the vectors are linearly independent.

2. Relevant equations

Triangle equality [itex] ||a+b|| \leq ||a||+||b|| [/itex]

[itex]||\alpha v|| =|\alpha|\cdot ||v||[/itex]

[itex] ||v|| = 0 -> \mathbf{v} = \mathbf{0}[/itex]

3. The attempt at a solution

I remember the definition of linear independence: [itex] k_1+v_1+k_2v_2\cdots k_Nv_N = 0[/itex] for non trivial vectors and all coefficients.

i use the triangle inequality and scalar multiplication:

[itex]||\sum\limits_{k=1}^Nc_k\mathbf{v}_k||^2 = \sum\limits_{k=1}^N|c_k|^2||\mathbf{v}_k||^2[/itex]

Which combined by the inequality stated in the problem implies that at least one of the vectors are different from the zero-vector

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# Homework Help: Linear independence in normed vector space

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