Linearly Dependent Vectors

n vectors X_1, X_2, ..., X_n are linearly dependent iff there exist scalars c_1, c_2, ..., c_n, not all zero, such that


If no such scalars exist, then the vectors are said to be linearly independent. In order to satisfy the criterion for linear dependence,

 c_1[x_(11); x_(21); |; x_(n1)]+c_2[x_(12); x_(22); |; x_(n2)]+...+c_n[x_(1n); x_(2n); |; x_(nn)]=[0; 0; |; 0]
 [x_(11) x_(12) ... x_(1n); x_(21) x_(22) ... x_(2n); | | ... |; x_(n1) x_(n2) ... x_(nn)][c_1; c_2; |; c_n]=[0; 0; |; 0].

In order for this matrix equation to have a nontrivial solution, the determinant must be 0, so the vectors are linearly dependent if

 |x_(11) x_(12) ... x_(1n); x_(21) x_(22) ... x_(2n); | | ... |; x_(n1) x_(n2) ... x_(nn)|=0,

and linearly independent otherwise.

Let p and q be n-dimensional vectors. Then the following three conditions are equivalent (Gray 1997).

1. p and q are linearly dependent.

2. |p·p p·q; q·p q·q|=0.

3. The 2×n matrix [p; q] has rank less than two.

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Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 272-273, 1997.

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Linearly Dependent Vectors

Cite this as:

Weisstein, Eric W. "Linearly Dependent Vectors." From MathWorld--A Wolfram Web Resource.

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