Show that for matrices 𝐴, 𝐵 ∈ ℝnxn, a vector 𝒘 = 𝐴𝐵𝒙 can be written as 𝒘 = ∑ni=1 (𝒃iT𝒙)𝒂i

Question

Show that for matrices 𝐴, 𝐵 ∈ ℝ^nxn, a vector 𝒘 = 𝐴𝐵𝒙 can be written as 𝒘 = ∑ⁿ_i=1 (𝒃_i^T𝒙)𝒂_i

where 𝒃𝑖^T is the 𝑖th row of matrix 𝐵, and 𝒂_i is the 𝑖th column of matrix 𝐴.

(Hint: Let 𝒛 = 𝐵𝒙, then 𝒘 = 𝑧₁𝒂₁ + 𝑧₁a₁+ ⋯ + 𝑧_n𝒂_n)

Answer 1

First, if z = Bx, then z_i = b_i^Tx for i = 1,2,...,n since b_i^T are the rows of B.

Second, Az = z₁a₁ + ... + z_na_n since a_i are the columns of A.

Therefore, w = ABx = (b₁^Tx)a₁ + ... + (b_n^Tx)a_n.