Linear Independence

What is linear independence?

Assume that we have a beautiful blue toy car with a plastic body and rubber tires. We may say that the toy car is constructed using some ratio of plastic to rubber. In other words, the material of the toy car is a linear combination of these. Moreover, we can say that the toy car is dependent on plastic and rubber, whereas it’s independent of wood because wood isn’t used in its construction.

The same concept can be extended to mathematical objects. If a mathematical object is made using a linear combination of others, then we call such an object linearly dependent on those objects. Otherwise, it’s independent of objects with linear combinations that aren’t used in its construction. Let’s write this linear independence concept formally.

Formal definition

An object, $\bold{z}$, is linearly independent of objects $\bold{x_1},\bold{x_2},\cdots,\bold{x_n}$ if and only if no choice of scalars $w_1,w_2,\cdots,w_n$ exists such that $\bold{z} = w_1\bold{x_1}+w_2\bold{x_2}+\cdots+w_n\bold{x_n}$.

Example: Axis of $xyz$ space

One way to represent the $x,y$, and $z$ axes in the standard $xyz$ space is $[$ ...