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.
$$ v=0 $$
