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Math Help - i'm puzzled about vectors&matrices..

  1. #1
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    Unhappy i'm puzzled about vectors&matrices..

    consider m vectors. You want to know if they're linear dependent. You can do that imposing their linear combination equal to zero and if you find the c1, c2, ...cn all equal to zero, then the vectors are independent. But there shoul be an easier way with determinants, but I am actually so confused, when detA=0 are the vectors dependent or independent?what about the rank? Can it help? I'm really puzzled...I know it would be too long to explain everything so if you could tell me a website where this things are explained in a straightforward way at least....thank you tons!
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  2. #2
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    The rows of a matrix are linerarly dependent if and only if the determinant of the matrix is zero. (The same goes for the columns).

    You probably know that if k is a number and A is a matrix, then k*det(A) is equal the the determinant of the matrix you get if you multiply one of the rows of A with k. Also, det(A) is the same as the determinant of the matrix you get if you take one row of A and add it to another. When the rows are linearly dependent you can use these facts to see that det(A)=(some number)*(the determinant of a matrix with all zeros on one of the rows)=0.
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