the difference between statistically independent and linearly independent? what is the difference between statistically independent and linearly independent concepts? 
 A: Consider a simple scenario in which you have two non-zero, non-constant, $n$-dimensional data vectors $\mathbf{X}$ and $\mathbf{Y}$.
They are linearly independent if there is no non-zero scalar $\alpha$ such that 
$\alpha \mathbf{X} - \mathbf{Y} = \mathbf{0}$
In other words, there is no non-zero multiplicative constant $\alpha$ that will transform $\mathbf{X}$ into $\mathbf{Y}$. Geometrically, this means that the vectors $\mathbf{X}$ and $\mathbf{Y}$ do not lie on the same line.
The two vectors $\mathbf{X}$ and $\mathbf{Y}$ are statistically independent if and only if their joint probability density is the product of their marginal probability densities, i.e.,
$f(\mathbf{X}, \mathbf{Y}) = f_X(\mathbf{X}) \cdot f_Y(\mathbf{Y})$
This implies 
cov$(\mathbf{X}, \mathbf{Y}) = \mathbf{0}$ 
(though the reverse implication is not true generally).
The two concepts are linked insofar as if the two vectors are not linearly independent then they can also not be statistically independent. For example, if for some non-zero scalar $\alpha$ we have
$\alpha \mathbf{X} = \mathbf{Y}$
then 
cov$(\mathbf{X}, \mathbf{Y}) = \text{cov}(\frac{1}{\alpha}\mathbf{Y}, \mathbf{Y}) = \frac{1}{\alpha} \text{var} (\mathbf{Y}) \ne \mathbf{0}$
However, linear independence of $\mathbf{X}$ and $\mathbf{Y}$ does not guarantee statistical independence (it is possible to have cov$(\mathbf{X}, \mathbf{Y}) \ne \mathbf{0}$ even if $\mathbf{X}$ and $\mathbf{Y}$ are linearly independent).    
