use determinant to decide if the set of vectors is linearly independent
How to determine if vectors are linearly independent using determinant?
We have now found a test for determining whether a given set of vectors is linearly independent: A set of n vectors of length n is linearly independent if the matrix with these vectors as columns has a non-zero determinant.
The set is of course dependent if the determinant is zero.How do you choose linearly independent vectors?
Given a set of vectors, you can determine if they are linearly independent by writing the vectors as the columns of the matrix A, and solving Ax = 0.
If there are any non-zero solutions, then the vectors are linearly dependent.
If the only solution is x = 0, then they are linearly independent.How do you know if a collection of vectors is linearly independent?
Really the simplest way to check if a set of vectors are linearly independent, is to put the vectors into a matrix, row reduce the matrix to echelon form, then the vectors are linearly independent if and only if there is a pivot in every column.
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