Proving parallelism: When the zero vector is a non-trivial linear combination of two vectors

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To prove that two vectors ( v ) and ( w ) are parallel if and only if the zero vector is a non-trivial linear combination of ( v ) and ( w ):

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Definitions:

1. Parallel Vectors: Two vectors ( v ) and ( w ) are parallel if there exists a scalar ( k ) such that ( v = k w ) or ( w = k v ).
2. Linear Combination: A vector ( u ) is a linear combination of ( v ) and ( w ) if ( u = \alpha v + \beta w ) for scalars ( \alpha ) and ( \beta ).
3. Non-Trivial Linear Combination: A linear combination is called non-trivial if ( \alpha ) and ( \beta ) are not both zero.
4. Zero Vector: The vector ( \mathbf{0} ) is represented as ( \mathbf{0} = 0 \cdot v + 0 \cdot w ), but for non-trivial combinations, ( \alpha \neq 0 ) or ( \beta \neq 0 ).

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Prove the “If” Direction

Assume the zero vector is a non-trivial linear combination of ( v ) and ( w ):

1. Let ( \alpha v + \beta w = \mathbf{0} ), where ( \alpha \neq 0 ) or ( \beta \neq 0 ).
2. Rearrange the equation:
   [
   \alpha v = -\beta w
   ]
3. Divide both sides by ( \beta ) (assuming ( \beta \neq 0 )) or ( \alpha ) (if ( \beta = 0 )):
   [
   v = -\frac{\beta}{\alpha} w
   ]
   This shows ( v ) is a scalar multiple of ( w ), hence ( v ) and ( w ) are parallel.

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Prove the “Only If” Direction

Assume ( v ) and ( w ) are parallel:

1. If ( v ) and ( w ) are parallel, there exists a scalar ( k ) such that ( v = k w ).
2. Substitute ( v = k w ) into the linear combination ( \alpha v + \beta w ):
   [
   \alpha v + \beta w = \alpha (k w) + \beta w = (\alpha k + \beta) w
   ]
3. To satisfy ( \alpha v + \beta w = \mathbf{0} ), we require ( \alpha k + \beta = 0 ).
   This means ( \beta = -\alpha k ), where ( \alpha \neq 0 ) or ( \beta \neq 0 ).
4. Therefore, the zero vector ( \mathbf{0} ) can be written as:
   [
   \mathbf{0} = \alpha v + \beta w
   ]
   where ( \alpha ) and ( \beta ) are not both zero, proving a non-trivial linear combination.

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Conclusion:

• If the zero vector is a non-trivial linear combination of ( v ) and ( w ), the two vectors are parallel.
• Conversely, if ( v ) and ( w ) are parallel, the zero vector can be expressed as a non-trivial linear combination of ( v ) and ( w ).

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Example:

1. Let ( v = (2, 4) ) and ( w = (1, 2) ).
   Clearly, ( v = 2 w ), so ( v ) and ( w ) are parallel.
2. A non-trivial linear combination yielding the zero vector is:
   [
   \mathbf{0} = 2(-1)(1, 2) + 1(2, 4)
   ]
   Here, ( \alpha = 2 ), ( \beta = -1 ), and ( \alpha, \beta \neq 0 ), confirming the proof.

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