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3. Numerical Linear Algebra 111

11. COMPUTATION OF TRANSFORMATION
MATRICES

If the vectors in the space1 are spanned by the independent vectors u1, u2,
u3, … un. Also if the vectors in the space2 are spanned by the independent
vectors v1, v2 …vn . They are called basis of the respective spaces.
Suppose the vector ‘u’ in the space 1 is mapped to the vector v in the
space 2.
(i.e.) T (u) = v. The vector ‘v’ can be represented as the linear
combinations of the independent vectors v1, v2, …vn.
Similarly the transformation of the basis vector u1 in the space1 can be
represented as the linear combinations of v1, v2, v3…vn.

Therefore T (u1) =a 11 v1+ a 12v2+ a 13 v3+ a 14v4 + … a 1n vn

Similarly T (u2) =a 21 v1+ a 22v2+ a 23 v3+ a 24v4 + … a 2n vn

…
T (un) =an 1 v1+ a n2v2+ a n3 v3+ a n4v4 + … a nn vn

The co-efficients of the vector v1, v2, v3 .. vn in the above mentioned
equation are arranged in the vector form and are called co-efficient vector as
given below.
The vector [a 11 a 12 a 13 a 14 … a 1n] is the co-efficient vector associated with
the vector T(u1) which is in the space 2. Similarly the vector
[a 21 a 22 a 23 a 24 … a 2n] is the co-efficient vector associated with the vector
T(u2).
Consider the vector u1 in the space 1 which can also be represented as
the linear combinations of the basis vectors u1, u2, u3, …un as given below.

u1=1*u1+0*u2+0*u3+….0*un.

In this case [1 0 0 0 …0] is the co-efficient vector associated with the
vector u1 in the space1.