Linear Dependence

Linear dependence is a fundamental concept in linear algebra, particularly relevant in the study of vector spaces. It describes a situation where a set of vectors can be expressed as a linear combination with non-zero weights that equals zero. This indicates that the vectors lie on a single line or plane, depending on the dimensionality of the space they occupy.

Definition and Explanation

A set of vectors is considered linearly dependent if there exists a set of weights, not all zero, such that their linear combination equals zero. Mathematically, for vectors ( \mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n ), they are linearly dependent if there exist scalars ( c_1, c_2, \ldots, c_n ), not all zero, such that:

[ c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2 + \cdots + c_n \mathbf{v}_n = \mathbf{0} ]

This condition implies that at least one of the vectors can be written as a linear combination of the others.

Geometric Interpretation

Geometrically, linear dependence can be visualized as points lying on a single line or plane. For instance, in a two-dimensional space, if vectors are linearly dependent, they lie on the same line. This is illustrated in Figure 2.11, which shows linearly dependent points in a 2D plane.

Figure 2.11 Linearly dependent points in a 2D plane

Example

Consider the row vectors ([1, 1]) and ([2, 2]). These vectors are linearly dependent because there exist non-zero weights, specifically (-2) and (1), such that:

[ -2[1, 1] + [2, 2] = [0, 0] ]

This example demonstrates that the second vector is a scalar multiple of the first, confirming their linear dependence.

Collinearity and Linear Dependence

Collinear vectors are a specific case of linear dependence. Vectors are collinear if they lie on a single straight line, which means they can be expressed as a linear combination with non-zero weights. This geometric property directly implies linear dependence, as collinear vectors satisfy the condition for linear dependence by definition.

In summary, linear dependence is a crucial concept in understanding the relationships between vectors in a vector space. It provides insight into the structure and dimensionality of the space, as well as the potential for redundancy among the vectors.