Express the vector \(\mathbf{v}=(1,2,3)\) as a linear combination of the vectors \(\mathbf{a}_{1}=(1,1,1), \mathbf{a}_{2}=(1,0,-2)\), and \(\mathbf{a}_{3}=(2,1,0)\)

A system of equations is a set of two or more equations with a common set of unknowns. In linear algebra, finding a vector as a linear combination of other vectors involves creating a system of equations that connects the coefficients of each vector.

For instance, when we need to express a given vector, denoted as \( \mathbf{v} \), as a combination of other vectors \( \mathbf{a}_{1}, \mathbf{a}_{2}, \mathbf{a}_{3} \), we set up an equation for each component of \( \mathbf{v} \) using the corresponding components of the vectors it's being expressed by. The solution to this system will provide us with the weights or coefficients (\( x_1, x_2, x_3 \)) that precisely balance the contribution of each vector to form \( \mathbf{v} \).

If the system has one solution, it means the vectors are linearly independent and span a space that includes \( \mathbf{v} \). If it has no solution, \( \mathbf{v} \) cannot be expressed as a linear combination of the given vectors, indicating that they do not span the required space.

The vector representation in linear algebra allows us to manipulate and combine vectors in meaningful ways. Vectors in \( \mathbb{R}^n \) space are represented as ordered lists of numbers, which are their components. Each vector can be thought of as an arrow pointing from the origin to a point in space.

When we talk about expressing one vector as a linear combination of others, we're essentially describing that target vector's position regarding the positions of the other vectors. The coefficients of the linear combination (the values that scale each vector) are crucial since they determine how far and in what direction we travel along each base vector to reach our destination vector. These coefficients are what we find by solving the system of equations.

In our exercise, vector \( \mathbf{v} \) was represented in the three-dimensional space as \( \mathbf{v} = (1, 2, 3) \), and the challenge was to find its expression in terms of other given vectors. This is invaluable in fields such as physics, engineering, and computer science where vector operations are foundational.

At the heart of linear algebra is the study of vectors and their properties. It's a branch of mathematics that facilitates the understanding of spaces that have both direction and magnitude. Linear algebra is fundamental in analyzing and manipulating these vectors and is the foundational theory behind many applications in science and engineering.

In solving our vector linear combination problem using linear algebra techniques, we've employed some crucial concepts such as linear independence, span, and basis vectors. These concepts tell us how vectors relate to each other and what kinds of spaces they can form through their combinations. Knowing how to work within these principles allows one to navigate vector spaces efficiently and solve related problems such as system of equations.

Furthermore, by applying matrix operations and vector space properties, linear algebra enables simplifying complex problems in several dimensions, making it an essential tool for many practical and theoretical applications.