Superposition principle If $S=(v_{1},v_{2}......v_{n})$ is a basis for vector Space V, then every vector v in V can be expressed in the form of $v=c_{1}v_{1}+.......c_{n}v_{n}$ in an unique way.
Explain the significance of this result in Quantum Mechanics.
The answer to the first one is easy as its proof is based on Linear algebra and I have already done it. However, I am struggling for the explanation of the second one, though I think that it is based on the superposition principle.
 A: It is an axiom in most formulations of QM that the state describing te system is a vector on Hilbet's space. So do not swet about it unless you wanna change the axioms (which will not be a bad idea)
A: The basis states are usually some orthogonal quantum states (for instance they can be the eigenstates of a Hermitian operator representing some observable). A general quantum state is then a superposition of these basis states. And because we are representing the quantum states by vectors in Hilbert space, the language of linear algebra is a natural language to descibe that.
A: So the result is that for any given vector v in V and for any given set $S = (v_1,v_2...)$ of basis states, there is a unique way of writing v in terms of the set S. The vector v is commonly used to represent a quantum state. The square amplitudes of the $v_i$ in v represent the probability of seeing $v_i$ when you do a measurement in the state represented by v: the Born rule. Suppose that the decomposition was not unique, then the square amplitudes might not be unique and so the state could not be used to predict probabilities since no single set of probabilities would follow from the state plus the Born rule. 
