Linear Dependence & Independence

Linear Dependence §

Linear Dependence indicates that a vector present in the set we are considering may be redundant, in other words we may represent the redundant vector as a linear combination of the other vector(s).

Linear Independence §

Linear independence indicates that no vectors in the set we are considering are redundant of the other one. This is to say that one may not be represented as a linear combination of another.

How do we determine LI or LD? §

A set of vectors $S$ is linearly dependent iff there exists a linear combination including a non-zero constant scalar and able to result in the zero-vector.

$0=-1v_1+c_2{v}_2+c_3{v}_3\dots c_n{v}_n$

From the above postulation we may determine LD & LI by setting the linear combination of the vectors we are considering equal to zero. If this is solvable, it is a Linear Dependent set. Otherwise, it is a Linear Independent set.

Example

A set of vectors is linearly independent if the only solution to the equation $c_1{v}_1+c_2{v}_2+c_3{v}_3=0$ is $c_1=c_2=c_3=0$.

#sapling