Linear Independence

Vectors $\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n$ are linearly independent if:

$$c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \cdots + c_n\mathbf{v}_n = \mathbf{0}$$

implies $c_1 = c_2 = \cdots = c_n = 0$.

What It Means

No vector can be written as a combination of the others. Each vector contributes something new.

When Vectors Are Dependent

• One vector is a combination of the others
• The matrix $A$ with these columns has $\det = 0$
• There’s a nontrivial solution to $Ax = 0$

In $\mathbb{R}^m$

• You can have at most $m$ independent vectors in $\mathbb{R}^m$
• More than $m$ vectors in $\mathbb{R}^m$ are always dependent