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I have 4 different vectors with 5 components. What is the fastest way to determine if these vectors form a linearly independent base in 5-dim space?

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4 vectors can't form a base in 5-dim space. You need at least 5 vectors
Of course... Well, how do I check if 5 vectors with 5 components form a linearly independet base? Let´s say it is not obvious and there is not a vector who looks like a multiple of another vector.
check determinant of matrix formed by this vectors. If it's equals 0 vectors are dependent. If not - independent.

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Oh, I did not know that I can use a determinant with vectors which are in a higher dim. than 3. Thx for your help myko!
you can do it in any dimention. The only thing you need is the matrix to be square

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