Let's consider an arbitrary 2x2 matrix as an example,
[tex]\mathbf A=\begin{bmatrix}\mathbf x&\mathbf y\end{bmatrix}=\begin{bmatrix}x_1&y_1\\x_2&y_2\end{bmatrix}[/tex]
The columns of [tex]\mathbf A[/tex] are linearly independent if and only if the column vectors [tex]\mathbf x,\mathbf y[/tex] are linearly independent.
This is the case if the only way we can make a linear combination of [tex]\mathbf x,\mathbf y[/tex] reduce to the zero vector is to multiply the vectors by 0; that is,
[tex]c_1\mathbf x+c_2\mathbf y=\mathbf 0[/tex]
only by letting [tex]c_1=c_2=0[/tex].
A more concrete example: suppose
[tex]\mathbf A=\begin{bmatrix}1&2\\4&8\end{bmatrix}[/tex]
Here, [tex]\mathbf x=\begin{bmatrix}1\\4\end{bmatrix}[/tex] and [tex]\amthbf y=\begin{bmatrix}2\\8\end{bmatrix}[/tex]. Notice that we can get the zero vector by taking [tex]c_1=-2[/tex] and [tex]c_2=1[/tex]:
[tex]-2\begin{bmatrix}1\\4\end{bmatrix}+\begin{bmatrix}2\\8\end{bmatrix}=\begin{bmatrix}-2+2\\-8+8\end{bmatrix}=\mathbf 0[/tex]
so the columns of [tex]\mathbf A[/tex] are not linearly independent, or linearly dependent.
