Step 1: Define linear operators. Linear operators satisfy two properties for any vectors \( X, Y \) and scalar \( \lambda \): \( A(X + Y) = AX + AY \) (additivity) \( (\lambda A)X = \lambda (AX) \) (scalar multiplication)
Step 2: Analyze each option. \( (A + B)X = AX + BX \): This is correct because the addition of two linear operators applied to \( X \) distributes over the addition. \( (\lambda A)X = \lambda (AX) \): This is correct because scalar multiplication of a linear operator applies directly to the result of \( AX \). \( (AB)X = A(BX) \): This is correct because the composition of two linear operators \( A \) and \( B \) acting on \( X \) satisfies this property. \( (A + B)X = A^T X + B^T X \): This is incorrect because the transpose (\( A^T \)) is not involved unless explicitly stated, and it does not apply to the given scenario.
Length of the streets, in km, are shown on the network. The minimum distance travelled by the sweeping machine for completing the job of sweeping all the streets is ________ km. (rounded off to nearest integer)