How do you convert coordinates from the standard basis to an alternative basis?

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 $\hat{j}$

A coordinate pair specifies the coefficients of a linear combination of the basis vectors. Without knowing the ordered basis, the numbers 3 and 2 do not tell you which vectors to scale and add.

Conditions: The vector space is equipped with a chosen ordered basis.; Coordinates are interpreted as coefficients of that basis.

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.