Linear Algebra 002 -- Changing basis

Changing Basis

• Basis transformation matrix: the basis vectors in my basis as columns 

 basis vectors are [3,1] and [1,1]

• transformation matrix * vector in new basis = vector in my basis

new basis vector: [3/2, 1/2], my basis vector: [5, 2]

• Moving the transformation matrix to the right gives us the vector in new basis

i.e. vector in my basis * inverse (transformation matrix)

 inverse (transformation matrix)

 inverse * vector in my basis

 result is vector in new basis

求inverse：

正对角线调换位置，副对角线反符号，整个➗determinant

Projection method

Only applicable when the new basis vectors are orthogonal.

• Find the dot product of transformation matrix and vector in new basis 

Example:

•  Transformation matrix in my basis
• Dot product the vector in my basis with each dimension of the transformation matrix to find the vector in new basis 

Vector Transformation in New Basis

 Formula: 

• Step 1: find out the vector in my basis

        ​​​​​​​

• Step 2: perform transformation in my basis

        

• Step 3: transform result back to new basis

        ​​​​​​​