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When a vector $$\mathbf{h}$$ is multiplied by a matrix $$\mathbf{W}_h$$ that is formed by the horizontal concatenation of several sub-matrices ($$\mathbf{W}_h^1, \mathbf{W}_h^2, \dots, \mathbf{W}_h^M$$), the operation is distributive. This means the result is equivalent to a new matrix formed by concatenating the products of the vector $$\mathbf{h}$$ with each individual sub-matrix. This property is mathematically expressed as: $$\mathbf{h} \mathbf{W}_h = \mathbf{h} \begin{bmatrix} \mathbf{W}_h^{1} & \mathbf{W}_h^{2} & \dots & \mathbf{W}_h^{M} \end{bmatrix} = \begin{bmatrix} \mathbf{h} \mathbf{W}_h^{1} & \mathbf{h} \mathbf{W}_h^{2} & \dots & \mathbf{h} \mathbf{W}_h^{M} \end{bmatrix}$$.

Vector Multiplication over Concatenated Matrices

True or False: When a row vector `h` is multiplied by a matrix `W` that is formed by horizontally concatenating two matrices `W_1` and `W_2` (i.e., `W = [W_1 W_2]`), the resulting vector is equivalent to the sum of the individual products `h * W_1` and `h * W_2`.

Based on the scenario described, explain why the engineer's proposed optimization produces the exact same final output vector as the original process. Describe the structure of the resulting vector from the single multiplication `h * W_h` in relation to the original, separate calculations.

Optimizing Matrix Computations

Consider a row vector `h = [2, 3]` and two matrices, `W_1 = [[1, 0], [0, 1]]` and `W_2 = [[4], [5]]`. If a larger matrix `W_h` is formed by horizontally concatenating `W_1` and `W_2` (i.e., `W_h = [W_1 W_2]`), what is the result of the multiplication `h * W_h`?

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