Linear Independence

Linear independence is a fundamental concept in linear algebra. It determines whether a set of vectors can be expressed as linear combinations of one another, which is critical for understanding vector spaces, basis, and dimension.

Definition

A set of vectors $\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n\}$ in a vector space is said to be linearly independent if the only solution to the equation:

$c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \dots + c_n\mathbf{v}_n = \mathbf{0}$

is:

$c_1 = c_2 = \dots = c_n = 0.$

If there exists a nontrivial solution (where some $c_i \neq 0$ ), the vectors are linearly dependent.

Geometric Interpretation

In $\mathbb{R}^2$ , two vectors are linearly independent if they do not lie on the same line through the origin.
In $\mathbb{R}^3$ , three vectors are linearly independent if they do not lie in the same plane through the origin.
More generally, a set of vectors is independent if no vector in the set can be expressed as a combination of the others.

Example

Consider the vectors:

$\mathbf{v}_1 = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}, \quad \mathbf{v}_2 = \begin{bmatrix} 4 \\ 5 \\ 6 \end{bmatrix}, \quad \mathbf{v}_3 = \begin{bmatrix} 7 \\ 8 \\ 9 \end{bmatrix}.$

To check for linear independence, solve:

$c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + c_3\mathbf{v}_3 = \mathbf{0}.$

In matrix form:

$\begin{bmatrix} 1 & 4 & 7 \\ 2 & 5 & 8 \\ 3 & 6 & 9 \end{bmatrix} \begin{bmatrix} c_1 \\ c_2 \\ c_3 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}.$

Performing row reduction reveals that the matrix has a rank of 2 (less than the number of vectors). Therefore, the vectors are linearly dependent.

Key Properties

A set containing the zero vector is always linearly dependent.
A single nonzero vector is always linearly independent.
The maximum number of linearly independent vectors in $\mathbb{R}^n$ is $n$ .
If a set of vectors is linearly dependent, at least one vector can be expressed as a linear combination of the others.

Applications

Linear independence is critical for several concepts in linear algebra:

Basis: A basis of a vector space is a linearly independent set that spans the entire space.
Dimension: The number of vectors in a basis defines the dimension of the space.
Matrix Rank: The rank of a matrix is the maximum number of linearly independent columns or rows.
Eigenvalues and Eigenvectors: Linear independence is used to find linearly independent eigenvectors associated with distinct eigenvalues.

Test for Linear Independence

To determine if a set of vectors is linearly independent:

Form a matrix with the vectors as columns.
Perform row reduction to obtain the reduced row echelon form (RREF).
Check the rank of the matrix:
- If the rank equals the number of vectors, they are linearly independent.
- If the rank is less than the number of vectors, they are linearly dependent.

Conclusion

Understanding linear independence is essential for working with vector spaces, solving systems of equations, and analyzing matrices. It provides the foundation for more advanced topics in linear algebra and its applications.