Linear Independence
For some sets of vectors v1,...,vn, a single vector can be written in two different ways as a linear combination of them:
Equivalently, by subtracting these a non-trivial combination is zero:
If that is possible, then v1,...,vn are called linearly dependent; otherwise, they are linearly independent. Similarly, we can speak of linear dependence or independence of an arbitrary set S of vectors.
If S is linearly independent and the span of S equals V, then S is a basis for V.
Read more about this topic: Linear Combination
Famous quotes containing the word independence:
“Children are as destined biologically to break away as we are, emotionally, to hold on and protect. But thinking independently comes of acting independently. It begins with a two-year-old doggedly pulling on flannel pajamas during a July heat wave and with parents accepting that the impulse is a good one. When we let go of these small tasks without anger or sorrow but with pleasure and pride we give each act of independence our blessing.”
—Cathy Rindner Tempelsman (20th century)