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Vn ) of E, let P = (ai j ) be the invertible matrix defined such that n vj = a i j ui , i=1 which is also the matrix of the identity id: E → E with respect to the bases (v 1 , . . , vn ) and (u1 , . . , un ), in that order (indeed, we express each id(vj ) = vj over the basis (u1 , . . , un )). The matrix P is called the change of basis matrix from (u1 , . . , un ) to (v1 , . . , vn ). Clearly, the change of basis matrix from (v1 , . . , vn ) to (u1 , . . , un ) is P −1 . Since P = (ai,j ) is the matrix of the identity id: E → E with respect to the bases (v1 , .

Vm ) be a basis for F . t. the bases (u1 , . . , un ) and (v1 , . . t. the dual bases (v1∗ , . . , vm ) and (u∗1 , . . , u∗n ) is the transpose M (f ) of M (f ). Proof . Recall that the entry ai j in row i and column j of M (f ) is the i-th coordinate of f (uj ) over the basis (v1 , . . , vm ). By definition of vi∗ , we have vi∗ , f (uj ) = ai j . The entry aj i in row j and column i of M (f ) is the j-th coordinate of f (vi∗ ) over the basis (u∗1 , . . , u∗n ), which is just f (vi∗ )(uj ) = f (vi∗ ), uj .

U∗n ) When a vector space E has infinite dimension, E and its bidual E ∗∗ are never isomorphic. When E is of finite dimension and (u1 , . . , un ) is a basis of E, in view of the canonical ∗∗ isomorphism cE : E → E ∗∗ , the basis (u∗∗ 1 , . . , un ) of the bidual is identified with (u1 , . . , un ). 27 can be reformulated very fruitfully in terms of pairings. 24 Given two vector spaces E and F over K, a pairing between E and F is a bilinear map −, − : E × F → K. Such a pairing is non-singular if for every u ∈ E, if u, v = 0 for all v ∈ F , then u = 0, and for every v ∈ F , if u, v = 0 for all u ∈ E, then v = 0.