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Dot Product: Matrix × Vector

Basics

A layer of neurons is just multiple dot products stacked: each row of the weight matrix dots with the input vector to produce one neuron's output. Here we compute each neuron manually, then show how np.dot does the whole layer at once. • Matrix columns must equal vector length. • Not commutative: np.dot(W, x) ≠ np.dot(x, W).

Each row of W dots with input x to produce one output. Row 1: (0.2×1)+(0.8×2)+(−0.5×3)+0.1 = 0.4, Row 2: (0.5×1)+(−0.91×2)+(0.26×3)−0.2 = −0.74, Row 3: (−0.26×1)+(−0.27×2)+(0.17×3)+0.05 = −0.24.

Python

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import numpy as np

x = np.array([1.0, 2.0, 3.0])
W = np.array([[ 0.2,   0.8,  -0.5 ],
              [ 0.5,  -0.91,  0.26],
              [-0.26, -0.27,  0.17]])
bias = np.array([0.1, -0.2, 0.05])

# ---- Without np.dot ----
n1 = (W[0,0]*x[0]) + (W[0,1]*x[1]) + (W[0,2]*x[2]) + bias[0]
n2 = (W[1,0]*x[0]) + (W[1,1]*x[1]) + (W[1,2]*x[2]) + bias[1]
n3 = (W[2,0]*x[0]) + (W[2,1]*x[1]) + (W[2,2]*x[2]) + bias[2]
print("Without np.dot:")
print("  Neuron 1: (0.2*1)+(0.8*2)+(-0.5*3)+0.1  =", round(n1, 2))
print("  Neuron 2: (0.5*1)+(-0.91*2)+(0.26*3)-0.2 =", round(n2, 2))
print("  Neuron 3: (-0.26*1)+(-0.27*2)+(0.17*3)+0.05 =", round(n3, 2))

# ---- With np.dot (one line!) ----
out = np.dot(W, x) + bias
print("\nWith np.dot:")
print("  np.dot(W, x) + bias =", out)

# ---- Not commutative: np.dot(x, W) would need compatible shapes ----
print("\nCommutativity:")
print("  np.dot(W, x) works: W is (3,3), x is (3,) -> output (3,)")
print("  np.dot(x, W) also works here (square W), but gives DIFFERENT result:")
print("  np.dot(W, x) =", np.dot(W, x))
print("  np.dot(x, W) =", np.dot(x, W))
print("  Not the same! Order matters for matrix-vector products.")

# For visualization
a = W
b = x.reshape(3, 1)

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Speed

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