Fill in the blanks in the following definitions A linear tra

Fill in the blanks in the following definitions.

A linear transformation F: V rightarrow W between vector spaces V and W is called an _____if there is a linear transformation G: W rightarrow V such that GF is the identity transformation on V and FG is the _____transformation on W. (b) Let V be a vector space with basis X = (v_1, ..., v_n) and W be a vector space with basis Y = (w_1, ..., w_m). Let F be a linear transformation from V to W. Then the matrix of F with respect to the bases X and Y, yFx, is the m times n matrix [fill in the matrix below] whose i-th column is the coordinate vector of _____ with respect to the basis _____.

Solution

(1) A linear transformation F:V->w between vector spaces V and W is called an INVERTIBLE TRANSFORMATION

if there exists a linear transformation G : W->V sucht that GF is the identity transformation on V and FG is the IDENTITY TRANSFORMATION on W.

(2) Let V be a vector space with basis X = (v[1],v[2],...v[n]) and W be a vector space with basis Y = (w[1],w[2],...w[n]) .Let F : V->W be a linear transformation from V to W. Then the matrix of F with respect to the bases X and Y is the mxn matrix

                                 a[11] a[12].......a[1n]

                                 a[21] a[22].......a[2n]

.                                ...................................

                                a[m1] a[m2].......a[mn]

whose kth column is the coordinate vector of F(v[k]) with respect to the basis Y of W..