By Stein M.R., Dennis R.K. (eds.)

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**Example text**

F (un ) . . a1n . . a2n .. . . . . amn The matrix M (f ) associated with the linear map f : E → F is called the matrix of f with respect to the bases (u1 , . . , un ) and (v1 , . . , vm ). When E = F and the basis (v1 , . . , vm ) is identical to the basis (u1 , . . , un ) of E, the matrix M (f ) associated with f : E → E (as above) is called the matrix of f with respect to the basis (u1 , . . , un ). 1, there is no reason to assume that the vectors in the bases (u1 , . . , un ) and (v1 , .

F (un )) spans E, and since E has dimension n, it is a basis of E (if (f (u1 ), . . 6. 9). Then, f is bijective, and by a previous observation its inverse is a linear map. We also have h = id ◦ h = (f −1 ◦ f ) ◦ h = f −1 ◦ (f ◦ h) = f −1 ◦ id = f −1 . This completes the proof. The set of all linear maps between two vector spaces E and F is denoted by Hom(E, F ) or by L(E; F ) (the notation L(E; F ) is usually reserved to the set of continuous linear maps, where E and F are normed vector spaces).

HAAR BASIS VECTORS AND A GLIMPSE AT WAVELETS We also find that the inverse of PV,U is −1 PV,U 1 0 0 0 1 1/3 0 0 = 1 2/3 1/3 0 . 1 1 1 1 Therefore, the coordinates of the polynomial 2x3 − x + 1 over the basis V are 1 1 0 0 0 1 2/3 1 1/3 0 0 −1 = 1/3 1 2/3 1/3 0 0 , 2 1 1 1 1 2 and so 2 1 2x3 − x + 1 = B03 (x) + B13 (x) + B23 (x) + 2B33 (x). 3 3 Our next example is the Haar wavelets, a fundamental tool in signal processing. 2 Haar Basis Vectors and a Glimpse at Wavelets We begin by considering Haar wavelets in R4 .