Introduction: The theorem we are going to prove is a "Goldilocks" theorem in linear algebra - it tells us that a basis is exactly the right size: not too small (it spans everything) and not too large (nothing is redundant).

Exercise: We must prove a biconditional statement `(P hArr Q)`:
- Forward (`rArr` , "Only If"): If a set `B = {beta_1 , ... \ , beta_n}` is a basis , then every vector `v in bbb"V"` has a unique expression `v = c_1 beta_1 + ... + c_n beta_n`.
- Backward (`lArr` , "If"): If every vector has a unique expression as a linear combination of vectors in `B` , then `B` must be a basis (it must span `bbb"V"` and be linearly independent).

General Strategy: For the Forward part , "existence" comes from the definition of a basis (spanning). To prove "uniqueness" , we use a classic math trick: assume there are two different ways to write the same vector , subtract them , and show they must actually be the same.

For the Backward part , "spanning" is given by the fact that every vector has an expression. "Linear independence" will be proven by looking at the specific case of the zero vector.

Specific Theory Used:
- Definition of Basis: A set that is both linearly independent and spans the space.
-Linear Independence: The only solution to `a_1 beta_1 + ... + a_n beta_n = bb{0}` is `a_1 = a_2 = ... = a_n = 0`.
- Spanning: Every vector in `bbb"V"` can be written as some linear combination of the set.

Solution:
`Part \ 1`: Basis `rArr` Unique Representation.
Assume `B = {beta_1 , ... \ , beta_n}` is a basis.
1. Existence: Since B is a basis , it spans `bbb"V"`. By definition , any `v in bbb"V"` can be written as `sum_(i = 1)^n c_i beta_i` for some scalars `c_i in bbb"F"`.
2. Uniqueness: Suppose there is another way to write v: `sum_(i = 1)^n k_i beta_i`.
3. Subtract two equations:
`bb{0} = v - v = sum_(i = 1)^n c_i beta_i - sum_(i = 1)^n k_i beta_i = sum_(i = 1)^n (c_i - k_i) beta_i`
4. Because `B` is a basis , it is linearly independent. This means the only way a sum of these vectors can be zero is if every coefficient is zero: `(c_i - k_i) = 0 implies c_i = k_i` for all i.
Thus , the two representations are identical.

`Part \ 2`: Unique Representation `rArr` Basis.
Assume every `v in bbb"V"` has a unique representation as `v = sum_(i = 1)^n c_i beta_i`.
1. Spanning: Since every vector has a representation , the set `B` spans `bbb"V"` by definition.
2. Linear Dependence: We look at the zero vector , and `bb{0} = 0*beta_1 + 0 * beta_2 + ... + 0*beta_n` is one way to write the zero vector.
3. Because the representation of every vector (including `bb{0}`) is unique , this must be the only way to write the zero vector.
4. Since the only solution to `sum c_i beta_i = bb{0}` is `c_i = 0` for all i , the set `B` is linearly independent.
5. Since `B` spans `bbb"V"` and is linearly independent , `B` is a basis. QED