Change of Basis

To translate a vector from one basis to another, simply multiply it by a matrix whose columns are the source basis vectors expressed in the target basis.

In other words:

Given:     
    A source basis S, consisting of the vectors s_1, s_2, ... s_n (expressed in the standard basis)
    A target basis T, consisting of the vectors t_1, t_2, ... t_n (expressed in the standard basis)
    A vector [v] _S (the vector to be translated, expressed in the source basis S)
    
(1) Express each source basis vector s_i in the basis T by solving:  
    s_i = k_1 t_1 +  k_2 t_2 + ... + k_n t_n
    [s_i] _T = (k_1, k_2, ... k_n)
       
(2) Create a transition matrix TM, where the columns are the source basis vectors expressed in the target basis, that is:
    c_i= [s_i] _T
       
(3) Multiply the original vector by the translation matrix to translate it to the target basis:
     [v] _T = [TM] [v] _S

Example 1:
    The vector {1/2,1/2} is given in the source basis {{1,0}, {1,1}}
    Translate it to the standard basis {{1,0}, {0,1}}

Here is the vector {1/2, 1/2} in the old basis {{1, 0}, {1, 1}}

[Graphics:HTMLFiles/CB1_27.gif]

*** Changing Basis ***

We have a vector {1/2, 1/2} expressed in the Source Basis {{1, 0}, {1, 1}}

We want to express the same vector in the Target Basis {{1, 0}, {0, 1}}

( {{1}, {0}} ) = k ( {{1}, {0}} ) + k ( {{0}, {1}} ) 1 2

Which yields the augmented matrix: ( {{1, 0, 1}, {0, 1, 0}} )

RREF of the matrix is: ( {{1, 0, 1}, {0, 1, 0}} ) , so k = 1 and k = 0 1 2

For the second source vector:

( {{1}, {1}} ) = k ( {{1}, {0}} ) + k ( {{0}, {1}} ) 1 2

Which yields the augmented matrix: ( {{1, 0, 1}, {0, 1, 1}} )

RREF of the matrix is: ( {{1, 0, 1}, {0, 1, 1}} ) , so k = 1 and k = 1 1 2

The source basis vectors expressed in terms of the Target Basis, which are {1, 0} and {1, 1} , become the columns of the Transition Matrix: ( {{1, 1}, {0, 1}} )

To find the vector of interest expressed in the Target Basis we multiply the original vector by the Transition Matrix:

( {{1/2}, {1/2}} ) ( {{1, 1}, {0, 1}} ) = {1, 1/2}

**********************

Here is the vector {1, 1/2} in the new basis {{1, 0}, {0, 1}}

[Graphics:HTMLFiles/CB1_44.gif]




Example 2:
    The vector {1/2^(1/2),1/2^(1/2)} is given in the source basis {{1/2^(1/2),1/2^(1/2)}, {-1/2^(1/2),1/2^(1/2)}}
    Translate it to the target basis {{3^(1/2)/2,1/2}, {-1/2,3^(1/2)/2}}

Here is the vector {1/2^(1/2), 1/2^(1/2)} in the old basis {{1/2^(1/2), 1/2^(1/2)}, {-1/2^(1/2), 1/2^(1/2)}}

[Graphics:HTMLFiles/CB1_56.gif]

*** Changing Basis ***

We have a vector {1/2^(1/2), 1/2^(1/2)} expressed in the Source Basis {{1/2^(1/2), 1/2^(1/2)}, {-1/2^(1/2), 1/2^(1/2)}}

We want to express the same vector in the Target Basis {{3^(1/2)/2, 1/2}, {-1/2, 3^(1/2)/2}}

( {{1/2^(1/2)}, {1/2^(1/2)}} ) = k ( {{3^(1/2)/2}, {1/2}} ) + k ( {{-1/2}, {3^(1/2)/2}} ) 1 2

Which yields the augmented matrix: ( {{3^(1/2)/2, -1/2, 1/2^(1/2)}, {1/2, 3^(1/2)/2, 1/2^(1/2)}} )

For the second source vector:

( {{-1/2^(1/2)}, {1/2^(1/2)}} ) = k ( {{3^(1/2)/2}, {1/2}} ) + k ( {{-1/2}, {3^(1/2)/2}} ) 1 2

Which yields the augmented matrix: ( {{3^(1/2)/2, -1/2, -1/2^(1/2)}, {1/2, 3^(1/2)/2, 1/2^(1/2)}} )

To find the vector of interest expressed in the Target Basis we multiply the original vector by the Transition Matrix:

( {{1/2^(1/2)}, {1/2^(1/2)}} ) ( {{1/4 (2^(1/2) + 6^(1/2)), 1/4 (2^(1/2) - 6^(1/2))}, {(-1 + 3^(1/2))/(2 2^(1/2)), 1/4 (2^(1/2) + 6^(1/2))}} ) = {1/2, 3^(1/2)/2}

**********************

Here is the vector {1/2, 3^(1/2)/2} in the new basis {{3^(1/2)/2, 1/2}, {-1/2, 3^(1/2)/2}}

[Graphics:HTMLFiles/CB1_73.gif]

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