## 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 .

**
Example:
**

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
.

**
Example:
**

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.

##
**
Note:
**

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 .

**
See Also
**
:
Basis Vector
,
Orthogonal Vector
,
Orthogonal Matrix

Rate this tutorial or give your comments about this tutorial