Testing Vectors for Linear Independence

A set of vectors is linearly independent if no vector in the set can be built as a combination of the others.

The Zero Combination

Linear independence asks whether any vector in a set is redundant. Formally, a collection of vectors is independent if the only way to sum them to zero is by setting every scalar coefficient to zero. If a non-trivial zero combination exists, at least one vector lies in the span of the rest.

c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2 + \dots + c_k \mathbf{v}_k = \mathbf{0} \implies c_1 = c_2 = \dots = c_k = 0

Matrix Null Space Connection

To check independence in R^n, stack the vectors as columns of a matrix A and solve A x = 0. If the homogeneous system has only the trivial solution x = 0, the columns are independent. Free variables in the echelon form reveal linear dependencies among the vectors.

A \mathbf{x} = \mathbf{0} \quad \text{has only } \mathbf{x} = \mathbf{0}

Geometric Interpretation

In two dimensions, two non-zero vectors are independent if and only if they are not parallel. In three dimensions, three vectors are independent if they do not all lie in the same flat plane through the origin. Higher dimensions generalize this notion of spanning new directions.

\dim(\text{span}(\mathbf{v}_1, \dots, \mathbf{v}_k)) = k

Flashcard

c_1 \mathbf{v}_1 + \dots + c_k \mathbf{v}_k = \mathbf{0}

Linear dependence relation defining redundant vectors in a set.

If there exist coefficients not all equal to zero that satisfy this equation, the set of vectors is linearly dependent. At least one vector can be expressed as a linear combination of the remaining vectors.

Check your understanding

Which of the following sets of vectors in R^3 is linearly independent?

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