A line has the equation $\displaystyle x_1 + 7x_2 = 0 $ in the standard basis of R2. The question is what is the equation in the basis given by the matrix $\displaystyle \left(\begin{array}{cc}2&3\\1&2\end{array}\right)$.

I'm used to doing this to vectors, so what I tried is to solve the equation $\displaystyle \left(\begin{array}{cc}2&3\\1&2\end{array}\right)\ left(\begin{array}{cc}y_1\\y_2\end{array}\right) = \left(\begin{array}{cc}1\\7\end{array}\right) $

The basis matrix is invertible, so $\displaystyle \left(\begin{array}{cc}2&-3\\-1&2\end{array}\right) \left(\begin{array}{cc}1\\7\end{array}\right) = \left(\begin{array}{cc}y_1\\y_2\end{array}\right)$

But that is all wrong. Apparently what I want to do is multiply with the transpose of the vector. Multiplying the vector to the transposed matrix also works. (So I guess my method would produce the right answer for an orthogonal matrix.)

But why? Why can't I take a vector to represent the normal of the line, change its basis and get the normal of the line represented in the new basis?

Thanks in advance for any help, and apologies if this is a too easy question that should go in some other section.