Lecture 5
Linear Dependence and Independence

The first of the two main concepts introduced in this course is the notion of a spanning set of vectors considered last time. The second will be considered here, and can be viewed as a description of whether a set of vectors is genuinely distinct (or more exactly, irredundant), given the allowable operations (scalar multiplication and addition).

Def. Let S = {u1 ,. . ., uk} be a nonempty set of vectors in Rn . S is called linearly dependent if there exist scalars c1 , . . ., ck , not all zero, such that

c1 u1 + · · · + ck uk = 0.
S is called linearly independent if it is not linearly dependent. Equivalently, this means that there are no scalars c1 , . . . , ck such that c1 u1 + · · · + ck uk = 0, except for the case where all of these scalars are zero.

Examples

Note: A collection of vectors is linearly dependent exactly when one of them can be expressed as a linear combination of the others, in other words, one of them is in the span of the others. That vector can be viewed as redundant, when using the collection to find its span.

How do we check whether a collection of vectors is linearly dependent?

Checking for Linear Dependence The set {u1 -, . . ., uk} in Rm is linearly dependent if, and only if, for A the matrix whose columns are given by these vectors, the equation
Ax = 0 has a nonzero solution.

Examples

We now characterize the condition which corresponds to the linear independence of a set of vectors, in terms of the matrix associated to it.

Theorem The following statements about an m x n matrix A are equivalent.
(a) The columns of A are linearly independent.
(b) The linear system Ax = b has at most one solution for each b in R m .
(c) The nullity of A is zero.
(d) The rank of A is n.
(e) The columns of the reduced row echelon form of A are distinct standard vectors in R m.
(f) The only solution to Ax = 0 is 0.

We have already seen that (a) and (f) are equivalent. By subtraction of two distinct solutions, it is easy to see that (b) and (f) are equivalent. Our earlier results show that (a) is equivalent to (c), (c) is equivalent to (d) by the definition of nullity, and (d) is easily seen to be equivalent to (e).

Note: the linear system Ax = b is called a homogeneous system if b = 0, and is called an inhomogeneous system otherwise.

Examples

Proposition
Any linear combination of solutions to a homogeneous linear system is again a solution of the same system.

The method we have for solving linear systems allows us to find the general solution of such a homogenous system, given in terms of free and determined variables.

Examples

The free variables are called "parameters," and the corresponding general solution is called a parametric representation. The general solution can be expressed as the span of a collection of vectors, with the free variables corresponding to the coefficients in the linear combinations of the spanning vectors.

Finally, we consider

Properties of Linearly Dependent or Independent Sets
(1) A set consisting of a single nonzero vector is linearly independent. On the other hand, any set containing the vector 0 is linearly dependent.
(2) A set consisting of a pair of vectors is linearly dependent if and only if one of the vectors is a multiple of the other. More generally, a set of two or more vectors is linearly dependent if and only if one of the vectors is a linear combination of the others.
(3) Let S be a linearly independent set in Rn, and let v be a vector in Rn. Then S È {v} is linearly independent if and only if v is not in the span of S.

Back 250 Lecture Index