Linear independence

Linear independence is a fundamental concept in linear algebra describing a family of vectors with no redundancy under linear combination. A family is linearly independent if the only way to combine its vectors to obtain the zero vector is to give every vector a zero coefficient. Otherwise, it is linearly dependent. The concept applies to elements of any vector space, including coordinate vectors, polynomials, matrices, and functions. (personal.math.ubc.ca)

Definition and scalar field

Let $V$ be a vector space over a field $F$. A finite family $v_1,\ldots,v_n$ is linearly independent when

implies $a_1=\cdots=a_n=0$. This is called the trivial solution. A solution with at least one nonzero coefficient is a linear dependence relation. Merely observing that zero coefficients satisfy the equation does not establish independence: they always do, even for dependent families. (opentext.uleth.ca)

For an infinite family, independence means that every finite subfamily is independent. Ordinary algebraic linear combinations contain only finitely many nonzero terms; the definition does not involve convergent infinite series. The empty family is independent because it admits no nontrivial relation. (personal.math.ubc.ca)

The scalar field matters. For example, $1$ and $i$, regarded as elements of the complex numbers, are independent over the real numbers: $a+bi=0$ for real $a,b$ forces both to vanish. Over the complex numbers they are dependent, since $i\cdot1-1\cdot i=0$. This follows directly from the definition with different permitted coefficients. (opentext.uleth.ca)

Redundancy and geometric examples

An equivalent criterion states that a family is dependent precisely when at least one vector is a combination of the others. Indeed, a nonzero coefficient in a dependence relation allows the corresponding vector to be isolated:

Removing that vector leaves the linear span unchanged. Dependence does not imply that every vector can be removed without changing the span. (personal.math.ubc.ca)

In $\mathbb R^2$, the vectors $(1,0)$ and $(0,1)$ are independent because

vanishes only when $a=b=0$. Adding $(1,1)$ creates the relation

In geometric terms, two nonzero vectors are independent exactly when they are not parallel. Three vectors in $\mathbb R^3$ are independent exactly when they do not all lie in a plane through the origin. (personal.math.ubc.ca)

A single nonzero vector forms an independent family. Any family containing the zero vector is dependent; an indexed family with a repeated vector is also dependent. Every subfamily of an independent family remains independent, while any family containing a dependent subfamily is dependent. (opentext.uleth.ca)

Bases, dimension, and uniqueness

An independent family forms a basis of its span, which is a linear subspace. Each vector in that span has a unique representation using the family: subtracting two representations would give a dependence relation, forcing corresponding coefficients to agree. Independence therefore guarantees uniqueness, whereas spanning guarantees existence. (opentext.uleth.ca)

In a vector space of finite dimension $d$, an independent family has at most $d$ members. An independent family of exactly $d$ vectors is a basis of the whole space. Smaller independent families can be extended to a basis by adding vectors outside their current span; finite spanning families can be reduced to a basis by removing redundant vectors. (ocw.mit.edu)

Matrix criteria and computation

Place coordinate vectors $v_1,\ldots,v_n\in F^m$ in the columns of a matrix $A$. Independence is equivalent to the homogeneous system of linear equations

having only the zero solution. Equivalently, the null space is $\{0\}$, the matrix rank is $n$, and row reduction produces a pivot in every column. Gaussian elimination can either establish independence or expose free variables yielding a dependence relation. In particular, $n>m$ guarantees dependence. (math.mit.edu)

These criteria also express a property of the linear map $T:F^n\to F^m$, $T(x)=Ax$: independent columns mean that $T$ is an injective function. Its kernel is zero. Thus independence concerns whether distinct coefficient vectors can produce the same output, not whether every vector in the target space is obtainable. (math.mit.edu)

Functions and numerical independence

For functions, a dependence relation must hold at every point of the domain. For example, the polynomials $1,x,x^2$ over $\mathbb R$ are independent, since the identity $a+bx+cx^2=0$ forces all coefficients to vanish. Evaluating functions at selected points can prove independence when the resulting evaluation columns are independent. Dependent evaluation columns alone, however, do not establish a relation valid throughout the domain. (personal.math.ubc.ca)

In numerical linear algebra, exact independence is distinguished from near dependence. Floating-point arithmetic can make theoretically zero quantities appear slightly nonzero. Rank tests based on singular value decomposition therefore compare singular values with a tolerance rather than requiring exact zeros. A numerically determined rank can depend on rounding error, data uncertainty, and the chosen threshold; it need not equal the exact algebraic rank of the stored or underlying matrix. (numpy.org)