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.
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.
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.
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.
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.
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.