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+.
Share on Twitter Share on Facebook
Comments
There are currently no comments
New Comment