Final answer:
Complex vectors are independent if there is no non-trivial combination of them that adds up to the zero vector. Two vectors with different magnitudes can add up to zero if they are in opposite directions. The commutativity of vector addition aids analysis but doesn't determine independence.
Step-by-step explanation:
In the context of complex vectors or any vectors in general, the concept of independence refers to whether the set of vectors can be expressed as a linear combination of one another. In the case of two complex vectors, they are independent if the only solution to the equation c1V1 + c2V2 = 0, where V1 and V2 are the vectors and c1 and c2 are complex scalars, is c1 = c2 = 0. If non-zero values for c1 and c2 exist that satisfy the equation, then the vectors are dependent.
For the vectors with different magnitudes, it is possible for two non-zero vectors to add to the zero vector if they are scalar multiples of each other but in opposite direction (meaning they are dependent). However, for three or more vectors it is possible for them to have different magnitudes and still sum to zero and be independent, although the conditions for this are more complex and depend on the dimensionality of the space.
The addition of vectors is indeed commutative, satisfying the equality A + B = B + A, which tells us about the properties of vector addition but does not directly inform us about independence. When working with vectors as in Equation 2.27, it is essential to express both vectors in their component forms to analyze their independence thoroughly.
If it is not known whether A and B are independent, it is a safe mathematical practice to assume they are dependent until proven otherwise. This principle safeguards against incorrect assumptions in vector analysis, particularly when solving linear equations or working with vector spaces.