❖ Linear Independence and Linear Dependence, Ex 1 ❖

To test for linear independence, write the coefficient equation as a homogeneous matrix system by placing the tested vectors into the columns of a matrix. Then perform row reduction on this matrix.

Conditions: You have a set of vectors to test.; You are forming a homogeneous system $Ax=0$.; The vectors are placed as columns in matrix $A$.

A set of vectors is linearly independent exactly when the only linear combination that equals the zero vector forces all coefficients to be zero. If there exists a nontrivial coefficient choice (at least one coefficient is nonzero) that produces the zero vector, the set is linearly dependent.

Conditions: The context is a set of vectors in a vector space.; We are testing for linear independence or dependence.

A family of nontrivial coefficients proves that the vectors are linearly dependent. Specifically, if you can find coefficients (not all zero) such that their linear combination with the vectors equals the zero vector, the set is dependent.

Conditions: You have a specific set of vectors.; You have found a parametric family of solutions to the homogeneous equation.; The parameter $k$ can be chosen to be nonzero.