A change of basis matrix is a matrix that represents the transformation of coordinates from one basis to another in linear algebra. It is used to convert vectors and matrices between different coordinate systems.
A change of basis matrix is important because it allows us to easily switch between different coordinate systems in linear algebra. It also helps us to solve problems and perform calculations in a more efficient manner.
To calculate a change of basis matrix, we need to follow a few steps. First, we need to determine the basis vectors of the original coordinate system and the new coordinate system. Then, we need to express the basis vectors of the new coordinate system in terms of the basis vectors of the original coordinate system. Finally, we can construct the change of basis matrix by arranging the new basis vectors as columns in the matrix.
A change of basis matrix and a linear transformation are closely related. The change of basis matrix represents the linear transformation that maps vectors from one coordinate system to another. In other words, the change of basis matrix is the matrix representation of the linear transformation between two different coordinate systems.
Yes, a change of basis matrix can be non-square. This can happen when the original and new coordinate systems have different dimensions. In this case, the change of basis matrix will have a different number of rows and columns, and it will not be a square matrix.