Question

How to determine if the columns of a matrix are linearly independent?

Answer 1

Let's consider an arbitrary 2x2 matrix as an example,

$\mathbf A=\begin{bmatrix}\mathbf x&\mathbf y\end{bmatrix}=\begin{bmatrix}x_1&y_1\\x_2&y_2\end{bmatrix}$

The columns of $\mathbf A$ are linearly independent if and only if the column vectors $\mathbf x,\mathbf y$ are linearly independent.

This is the case if the only way we can make a linear combination of $\mathbf x,\mathbf y$ reduce to the zero vector is to multiply the vectors by 0; that is,

$c_1\mathbf x+c_2\mathbf y=\mathbf 0$

only by letting $c_1=c_2=0$.

A more concrete example: suppose

$\mathbf A=\begin{bmatrix}1&2\\4&8\end{bmatrix}$

Here, $\mathbf x=\begin{bmatrix}1\\4\end{bmatrix}$ and $\amthbf y=\begin{bmatrix}2\\8\end{bmatrix}$. Notice that we can get the zero vector by taking $c_1=-2$ and $c_2=1$:

$-2\begin{bmatrix}1\\4\end{bmatrix}+\begin{bmatrix}2\\8\end{bmatrix}=\begin{bmatrix}-2+2\\-8+8\end{bmatrix}=\mathbf 0$

so the columns of $\mathbf A$ are not linearly independent, or linearly dependent.