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,...,v_m ∈ V. Concurrently, I identified a set of scalars,α₁,...,α_m ∈ 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.

$$v=0$$