Representing Compositions of Linear Transformations as Matrices

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Let S:U->V and T:V->W be linear transformations, and let A = [T][α,β] and B = [S][β,γ] with respective bases of U, V, and W given as α = {u_1,..,u_n}, β = {v_1,..,v_m}, and γ = {w_1,..,w_p}; [1 < n,m,p < ∞] ∈ Z+. The product of these two matrices AB = [TS][α,γ].

Then we define T(v_k) and S(u_j) as follows:

T(v_k) = ∑(i=1, p)(A_ik * w_i), and S(u_j) = ∑(k=1, m)(B_kj * v_k).

[TS][α,γ] is given as follows:

TS(u_j) = T(∑(k=1, m)(B_kj * v_k)) 
= ∑(k=1, m)(B_kj) * T(v_k) 
= ∑(k=1, m)(B_kj) * ∑(i=1, p)(A_ik * w_i) 
= ∑(i=1, p)(∑(k=1, m)(A_ik * B_kj))(w_i) 
= ∑(i=1, p)(C_ij * w_i),

where C_ij = ∑(k=1, m)(A_ik * B_kj).

This result motivates the more general definition of matrix multiplication, which states that the product of A and B is given as follows:

(AB)_ij =  ∑(k=1, n)(A_ik * B_kj); [1 ≤ i ≤ m, 1 ≤ j ≤ p] ∈ Z+.

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