Column1 = Temp_F and Column2 = Temp_C -- are these linearly dependent? If $X$ is a matrix of size $m$ x $2$, where the first column is a range of Celsius values and the second column is their corresponding Fahrenheit values, would the columns be considered linearly dependent? 
In a $2$ x $2$ matrix (of boiling and freezing points)
$$X=
\begin{bmatrix}
  0 &32  \\
100 &212
\end{bmatrix}$$
the determinant is not zero.
However, given that $C = (F - 32) / 1.8$, it seems that you could create a matrix where where a linear combination of some columns are equal to some other column.
So, in the first example -- the matrix is non-singular, whereas in the second it is?
 A: In a Wikipedia article I found this:

A set of vectors is said to be linearly dependent if one of the vectors in the set can be defined as a linear combination of the other vectors. If no vector in the set can be written in this way, then the vectors are said to be linearly independent.

If you have only two vectors $x_1$ and $x_2$, they will be linearly dependent if and only if $x_1=c \cdot x_2$ for some $c \neq 0$. 
Clearly, the temperature in Fahrenheit and the temperature in Celsius does not satisfy this condition, so the vectors are linearly independent. Meanwhile, if you had a third vector -- a column of ones (or a column of any constant, for that matter) -- the three vector system would be linearly dependent as the temperature in Fahrenheit is a linear combination of a unit (or any constant) and the temperature in Celsius.
In a regression setting, you can use both the temperature in Fahrenheit and the temperature in Celsius as regessors if you do not include intercept. However, if you do include intercept, you can use only one of them. Meanwhile, including both of the temperatures and the intercept would lead to perfect multicollinearity and noninvertibility of the $X'X$ matrix. 
I think it makes more sense to include only one of them and an intercept as the interpretation will be way more straightforward. On the other hand, including both the temperature in Fahrenheit and the temperature in Celsius without the intercept seems like an unnecessarily complicated, nontransparent way to estimate the intercept and find out the effects of temperature.
