Vectors Property: Necessary & Sufficient Conditions

[tc]

Consider N vectors, V_1 ... V_N, (not all zero) in an N-dimensional space. Project these vectors to a 2-dimensional plane, P. Let us denote the results for the i'th vector as (a_i,b_i) in Cartesian coordinates defined on the plane. We will be interested in considering these coordinates as a complex number: x_i = a_i + j*b_i.

Now, these N vectors hold the following property: regardless of the chosen P, the sum of all {x_i}^2 is always equal to 0.

The question: give an explicit description of the property that the N vectors most hold. Prove that this property is both sufficient and necessary.

 

[matt grime]

You should at least say wrt to what basis you're writing the x_i

 

[M98Ranger]

The property that the N vectors must hold is that they are linearly dependent. This means that at least one of the vectors can be written as a linear combination of the remaining vectors. In other words, there exists a set of coefficients such that when each vector is multiplied by its corresponding coefficient and added together, the result is the zero vector (a vector with all components equal to 0).

To prove that this property is both sufficient and necessary, we will first show that it is sufficient. This means that if the N vectors hold this property, then the sum of all {x_i}^2 will always be equal to 0, regardless of the chosen plane P.

Assume that the N vectors are linearly dependent. This means that there exists a set of coefficients c_1, c_2, ..., c_N (not all zero) such that:

c_1*V_1 + c_2*V_2 + ... + c_N*V_N = 0

Now, let us consider the projection of these vectors onto a 2-dimensional plane P. This projection can be represented by the complex numbers x_i = a_i + jb_i for each vector V_i. Since the vectors are linearly dependent, we can rewrite the above equation as:

c_1*x_1 + c_2*x_2 + ... + c_N*x_N = 0

Expanding this equation, we get:

c_1*(a_1 + jb_1) + c_2*(a_2 + jb_2) + ... + c_N*(a_N + jb_N) = 0

Rearranging terms, we get:

(c_1*a_1 + c_2*a_2 + ... + c_N*a_N) + j(c_1*b_1 + c_2*b_2 + ... + c_N*b_N) = 0

Since the real and imaginary parts of a complex number must both be equal to 0 for the number to be equal to 0, we can conclude that:

c_1*a_1 + c_2*a_2 + ... + c_N*a_N = 0

and

c_1*b_1 + c_2*b_2 + ... + c_N*b_N = 0

But these equations are exactly the conditions for the sum of all {x_i}^2 to be equal to 0. Therefore, we