Understanding Linear Dependence and Independence

In the study of vectors and linear algebra, the concepts of linear dependence and independence are fundamental. They describe the relationships between vectors in a vector space and have important implications for solving systems of linear equations, transforming spaces, and more.

Definitions

Linear Dependence

A set of vectors is said to be linearly dependent if at least one of the vectors in the set can be expressed as a linear combination of the others. In other words, there exist coefficients (not all zero) such that a linear combination of the vectors equals the zero vector.

Mathematically, a set of vectors ${\vec{v}_1, \vec{v}_2, ..., \vec{v}_n}$ in a vector space $V$ is linearly dependent if there exist scalars $c_1, c_2, ..., c_n$, not all zero, such that:

$$ c_1\vec{v}_1 + c_2\vec{v}_2 + ... + c_n\vec{v}_n = \vec{0} $$

Linear Independence

Conversely, a set of vectors is linearly independent if no vector in the set can be written as a linear combination of the others. For a set of vectors to be linearly independent, the only solution to the equation above must be the trivial solution where all coefficients are zero.

A set of vectors ${\vec{v}_1, \vec{v}_2, ..., \vec{v}_n}$ in a vector space $V$ is linearly independent if the only scalars $c_1, c_2, ..., c_n$ that satisfy

$$ c_1\vec{v}_1 + c_2\vec{v}_2 + ... + c_n\vec{v}_n = \vec{0} $$

are $c_1 = c_2 = ... = c_n = 0$.

Table of Differences

Aspect	Linearly Dependent	Linearly Independent
Definition	At least one vector is a linear combination of others.	No vector can be expressed as a linear combination of others.
Coefficients	There exist coefficients, not all zero, that satisfy the linear combination.	The only coefficients that satisfy the linear combination are all zero.
Zero Vector Equation	$c_1\vec{v}_1 + ... + c_n\vec{v}_n = \vec{0}$ has a non-trivial solution.	$c_1\vec{v}_1 + ... + c_n\vec{v}_n = \vec{0}$ has only the trivial solution.
Basis	Cannot form a basis for a vector space.	Can form a basis for a vector space.
Dimension	The set contains more vectors than the dimension of the space (if finite).	The maximum number of linearly independent vectors equals the dimension of the space (if finite).
Implications	Redundancy in the set of vectors; some vectors are "extra."	Each vector adds a new dimension to the spanned space; no redundancy.

Examples

Example 1: Linear Dependence in $\mathbb{R}^2$

Consider the vectors $\vec{v}_1 = \begin{bmatrix} 1 \ 2 \end{bmatrix}$ and $\vec{v}_2 = \begin{bmatrix} 2 \ 4 \end{bmatrix}$ in $\mathbb{R}^2$. We can see that $\vec{v}_2 = 2\vec{v}_1$, so these vectors are linearly dependent. The equation $c_1\vec{v}_1 + c_2\vec{v}_2 = \vec{0}$ has non-trivial solutions, such as $c_1 = 2$ and $c_2 = -1$.

Example 2: Linear Independence in $\mathbb{R}^3$

Consider the vectors $\vec{v}_1 = \begin{bmatrix} 1 \ 0 \ 0 \end{bmatrix}$, $\vec{v}_2 = \begin{bmatrix} 0 \ 1 \ 0 \end{bmatrix}$, and $\vec{v}_3 = \begin{bmatrix} 0 \ 0 \ 1 \end{bmatrix}$ in $\mathbb{R}^3$. These vectors are linearly independent because the only solution to $c_1\vec{v}_1 + c_2\vec{v}_2 + c_3\vec{v}_3 = \vec{0}$ is $c_1 = c_2 = c_3 = 0$. They also form a basis for $\mathbb{R}^3$.

Conclusion

Understanding linear dependence and independence is crucial for various applications in mathematics and engineering. Linearly independent sets of vectors can span vector spaces and form bases, which are essential for coordinate transformations, solving systems of equations, and more. Recognizing whether a set of vectors is linearly dependent or independent is a key skill in linear algebra.