# Dimensional Analysis: Buckingham Pi Theorem Using a Matrix

I have the following problem - which is basically that $\omega = f(g,h,l)$ now the book claims that the two $\pi$ terms as follows $\pi_1 = \omega \sqrt{\frac{l}{g}}$ and $\pi_2 = \frac{h}{l}$. Now we know we can get different dimensional $\pi$ terms because of the nature of the theorem. The ones I get however I obtain by solving $Ax=0$ for a matrix $A$ so my solution space looks like this $x = \begin{pmatrix} a \\ b \\ c \\ d \end{pmatrix} = c \begin{pmatrix} 2 \\ -1 \\ 1 \\ 0 \end{pmatrix} + d \begin{pmatrix} 2 \\ -1 \\ 0 \\ 1 \end{pmatrix}$ such that my $\pi$ terms are then $\pi_1 = \frac{\omega^2 h}{g}$ and $\pi_2 = \frac{\omega^2 l}{g}$ However I am uncertain as to whether or not this is the correct answer. It seems right to me given the theorem, but I don't see any way to confirm that.

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Your two vectors work fine, but their difference is (0,0,1,-1) which gives the book's term h/l (or l/h depending on your convention). If x can be written as a sum of two vectors A and B, it can also be written as a sum of A and A-B, since $aA+bB = (a+b)A + b(B-A)$.