Linear Combination of Vectors

A linear combination is the sum of products between vectors and their corresponding scalars.

To provide some context, consider a vector space, V, defined over a field K. In one of my explorations, I selected m vectors from this space, denoted as v1,...,vm ∈ V. Concurrently, I identified a set of scalars,α1,...,αm ∈ K from K. The linear combination of these vectors and scalars is: $$ v=α_1 v_1 + ... + α_m v_m $$

What remains compelling to me is that this resultant vector still firmly resides within V. This is attributed to the foundational operations employed - addition and multiplication - which consistently produce results intrinsic to V.

Example

For clarity, I'll elucidate with an example.

Within the confines of the R3 field for a vector space, V, I once delineated two vectors:

$$ v_1 = \{ 4,5,6 \} $$ $$ v_2 = \{ 7,8,9 \} $$

For this particular study, I also incorporated two scalars from the real number domain:

$$ α_1=1 , α_2=2 $$

The linear combination of these vectors, facilitated by the scalars, was derived by multiplying each vector by its scalar and subsequently aggregating the results:

$$ v = α_1 v_1 + α_2 v_2 $$

Sequentially, this evolved as:

$$ v = 1 \cdot v_1 + 2 \cdot v_2 $$

$$ v = 1 \cdot \{ 4,5,6 \} + 2 \cdot \{ 7,8,9 \} $$

$$ v = \{ 4,5,6 \} + \{ 14,16,18 \} $$

$$ v = \{ 4+14,5+16,6+18 \} $$

$$ v = \{ 18,21,24 \} $$

Upon conclusion, I discerned that the vector \{ 18,21,24 \} was the culmination of the vectors and scalars I had initially chosen.

Trivial linear combination

However, a pertinent consideration arose: What if all the scalars were zero? This scenario ushers in the concept of the trivial linear combination:

$$ v= 0 \cdot v_1 + ... + 0 \cdot v_m $$

Herein:

$$ \{ α_1 , ... , α_m \} = \{ 0 , ... , 0 \} $$

The inevitable conclusion? The vector, v, invariably equates to zero, synonymous with a null vector.

$$ v=0 $$

 
 

Please feel free to point out any errors or typos, or share suggestions to improve these notes. English isn't my first language, so if you notice any mistakes, let me know, and I'll be sure to fix them.

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