Why does the coordinate pair (3,2) only make sense together with its chosen ordered basis?

Use the inverse of the change-of-basis matrix $P$. If $P$ converts from alternative to standard coordinates, then $P^{-1}$ converts from standard to alternative coordinates.

Conditions: The change-of-basis matrix $P$ is invertible.; The input vector is given in standard coordinates.

In linear algebra, a coordinate pair like $(3, -2)$ is viewed not just as a static location, but as scaling factors. The first number scales the horizontal unit vector $\hat{i}$, and the second scales the vertical unit vector $\hat{j}$.

Conditions: Working within a standard 2D Cartesian coordinate system; Using the standard unit basis vectors $\hat{i}$ and $\hat{j}$

To form a basis, a set of vectors must be both linearly independent and span the target space. Linear independence means each vector contributes a brand-new dimension that cannot be recreated by the others.

Conditions: The vectors belong to the target vector space; The set is minimal (no redundant vectors)

Construct a matrix $P$ whose columns are the alternative basis vectors expressed in standard coordinates. Multiplying $P$ by a coordinate vector in the alternative basis yields the standard coordinates of that vector.

Conditions: The alternative basis vectors are known in standard coordinates.; The input vector is given as coordinates in the alternative basis.

To represent a standard-coordinate transformation $A$ in a new basis, use the formula $P^{-1}AP$. First convert the input vector to the new basis using $P^{-1}$, apply the transformation $A$ in standard coordinates, then convert the result back to the new basis using $P$.

Conditions: The transformation $A$ is defined in standard coordinates.; The change-of-basis matrix $P$ is invertible.