2.4 Linear independence
In this section, our focus turns to the uniqueness of solutions of a linear system, the second of our two fundamental questions asked in Question 1.4.2. This will lead us to the concept of linear independence.
Reading
Try out the Preview Activity and read section Chapter 2 Section 4 in Understanding Linear Algebra by Davis Austin.
Definition 1: A set of vectors [latex]\{\vec{v}_1,\vec{v}_2,\dots,\vec{v}_n \}[/latex] is called linearly dependent if one of the vectors is a linear combination of the others. Otherwise, the set of vectors is called linearly independent.
Definition 2: A set of vectors [latex]\{\vec{v}_1,\vec{v}_2,\dots,\vec{v}_n \}[/latex] is called linearly dependent if [latex]c_1\vec{v}_1+\dots+c_n\vec{v}_n =\vec{0}[/latex] has more than one solution. Otherwise, if the only solution is [latex]c_1=0, c_2=0, \dots, c_n=0[/latex] then the set of vectors is called linearly independent.
Proposition 2.4.2.
The columns of a matrix are linearly independent if and only if in its reduced row echelon form, every column contains a pivot.