By Andrew Baker
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N), i=1 n wk = b1k v1 + · · · + bmk wm = bik wi i=1 (k = 1, . . , m). 40 3. LINEAR TRANSFORMATIONS Working with these bases we obtain the matrix T We will set t ··· 11 .. = [tij ] = . tm1 · · · [f ]S a11 · · · . A = [aij ] = .. an1 · · · a1n .. . , ann t1n .. . . tmn b11 · · · . B = [bij ] = .. bm1 · · · b1m .. . . 35. (a) Let x ∈ V have coordinates x1 , . . , xn with respect to the basis S : v1 , . . , vn of V and let f (x) ∈ W have coordinates y1 , .
Wm , 0, . . 30 to show that there is a unique linear transformation f : V −→ W with the stated properties. 29. The next result provides a convenient way to decide if two vector spaces are isomorphic: simply show that they have the same dimension. 33. Let V and W be finite dimensional vector spaces over the field F . Then there is an isomorphism V −→ W if and only if dimF V = dimF W . Proof. 21(c). 32(e) there is an isomorphism V −→ W . Now we will see how to make use of bases for the domain and codomain to work with a linear transformation by using matrices.
The initial case is n = 2 where det(E(R1 ↔ R2 )) = 0 1 = −1. 1 0 If R = Rr → λRr for λ = 0, then expanding along the r-th row gives det(E(Rr → λRr )) = λ det(In ) = λ. Finally, if R = Rr → Rr + λRs with r = s, then expanding along the r-th row gives det(E(Rr → Rr + λRs )) = det(In−1 ) + λ det(Inrs ) = 1 + λ × 0 = 1. Thus to calculate a determinant, first find any sequence R1 , . . Rk of elementary row operations so that the combined effect of applying these successively to A is an upper triangular matrix L.