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.
The respective values of \( |\vec{a}| \) and} \( |\vec{b}| \), if given \[ (\vec{a} - \vec{b}) \cdot (\vec{a} + \vec{b}) = 512 \quad \text{and} \quad |\vec{a}| = 3 |\vec{b}|, \] are:
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)