Page 185 - A Course in Linear Algebra with Applications

6.2: Linear Transformations and Matrices 169

Change of basis
Being aware of a dependence on the choice of bases, we
wish to determine the effect on the matrix representing a linear
transformation when the ordered bases are changed. The first
step is to find a matrix that describes the change of basis.
Let B = {v 1,..., v n } and B' = {v^,..., v' n} be two or-
dered bases of a finite-dimensional vector space V. Then each
v^ can be expressed as a linear combination of i , . . . , v n , say
v

n

J'=l
for certain scalars Sji. The change of basis B' —> B is deter-
mined by the n x n matrix S = [sij]. To see how this works
we take an arbitrary vector v in V and write it in the form

n

i=l
where, of course, c\ ,..., c n' are the entries of the coordinate
vector [v]g/. Replace each v / by its expression in terms of the
Vj to get

n n n n

i = l j = X j = l i = l
From this one sees that the entries of the coordinate vector
s c
[V]B are just the scalars Y17-1 ji 'n ^ or 3 = 1, 2,..., n. But the
latter are the entries of the product

(c'A

Vn)