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I want to whether there is a function (/matrix) such that it can generate a m-dimensional vector such that this generated vector will always be linearly independent of the set of vectors the function has already generated.

My problem can be written in pseudocode format as follow. I therefore expect that any m randomly picked vectors from the pool of the N vectors will generate a full-rank matrix.

For (n=1; n<N; n++) { //N>m

S = Span (v_{1}, v_{2}, ..., v_{n-1})

Generate vector v_{n}, such that v_{n}is not an element of S; //i.e. v_n is linearly independent of the set of vectors already generated.

S = Span (v_{1}, v_{2}, ..., v_{n})

}

Vandermonde matrix is one possible option, but it requires the use of exponentially large field size. So I am looking for vectors generated over smaller field size. Any help in this direction will be greatly appreciated.

Thanks in advance.

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# Function to generate linearly independent vectors

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