Change of Basis
In this page you will learn how you can transform a point from one coordinate system into another coordinate system.
In the previous topic of Basis Vector , you have learned that we can change the coordinate system. Suppose you have a point with coordinate of in Euclidean space (with standard basis coordinate and ). We want to find the coordinate of the same point in a new coordinate system form by basis vector and . The figure below shows that the new coordinate of the same point is . In this page, I will show you how you will obtain the coordinate in the new coordinate system.
Suppose we start with a point in the standard Euclidean basis and . We want to transform it into a new space span by basis vectors and . First, we can do horizontal concatenation of the new basis vectors into a matrix . Then, the coordinate of a point in the old basis is equal to the matrix multiplication of the augmented matrix of the new basis with coordinate of the point in the new basis . That is . Thus to get the coordinate of a point in the new basis is the reverse, that is .
Our point in the Euclidean basis is . Our new basis is vector and . Augmenting the basis vectors form a matrix . The inverse of the matrix is . The coordinate of the point in the new coordinate is .
Now suppose we want to find back the coordinate of point from the coordinate system of basis vectors and into Euclidean system. Augmenting the basis vectors form a matrix . We have our coordinate point back.
Transformation of coordinate systems follows equality of matrix-vector multiplication , where and are the matrix of the respective basis vectors. In Euclidean coordinate system, the basis vectors form identity matrix . Thus, the formula can be simplified into or .