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3. Numerical Linear Algebra 115

11.2 Basis Co-efficient Transformation

Consider the 4-dimensional vector space which are spanned by the basis
T
T
T
T
u1=[1 0 0 0] , u2=[0 1 0 0] and u3=[0 0 1 0] and u4=[0 0 0 1] . Consider
another set of orthonormal basis which spans the space.

T
v1 = (1/2)* [1 1 1 1 ]

T
v2 = (1/2) * [1.0000 0 - 1.0000i -1.0000 0 + 1.0000i ]

T
v3 = (1/2) * [1 -1 1 -1]

v4 = (1/2)* [1.0000 0 + 1.0000i -1.0000 0 - 1.0000i ] T

Any vector in the space can be represented as the linear combination of
u1, u2, u3 and u4. The same vector in the space can be represented as the
linear combination of v1, v2, v3 and v4.
Consider the transformation T which transforms the vector v is
represented as T(v) and is equal to v.

v is represented as the linear combination of u1 u2 u3 and u4.Let the co-
efficient vector be [p1 p2 p3 p4]

T(v)=v is represented as the linear combination of v1,v2,v3,v4.Let the
co-efficient vector be [q1 q2 q3 q4].

The transformation matrix which converts the co-efficient vector
[p1 p2 p3 p4] into the co-efficient vector [q1 q2 q3 q4] is transformation
matrix for the change of basis. It is obtained as described below.

T([1 0 0 0])=[1 0 0 0] is represented as 0.5*v1+0.5*v2+0.5*v3+0.5*v4

T([0 1 0 0])=[0 1 0 0] is represented as
0.5*v1 + ( 0.5 i )*v2+ (-0.5)*v3+ (-0.5 i)*v4

T([0 0 1 0])=[0 0 1 0] is represented as
0.5*v1 + (- 0.5)*v2+ (0.5)*v3+ (- 0.5 ) *v4

T([0 0 0 1])=[0 0 0 1] is represented as
0.5*v1 + (- 0.5 i)*v2+ (- 0.5)*v3+ ( 0.5 i) *v4