91Ó°ÊÓ

Base vectors are essential in understanding vector coordinates in a given space. When we discuss base vectors, we're referring to a set of vectors that define a coordinate system in a vector space. For example, in the exercise given, the basis \( S = \{ (1, 1), (2, 3) \} \) is defined for the two-dimensional space \( \mathbf{R}^{2} \). This means that any vector in the space can be expressed as a linear combination of these base vectors.

To understand this better, picture a grid in a two-dimensional plane. Each base vector represents a direction and a unit of measurement along that grid. In essence, base vectors are the building blocks that allow us to describe any other vector within the space by scaling and adding them appropriately. When you have two vectors in \( \mathbf{R}^{2} \), if they are not parallel, they can serve as a basis for the plane.

A linear combination in vector mathematics involves creating a new vector using scalar multiples of other vectors. The scalars are real numbers that you multiply your vectors by before adding them together. If you're working with the basis vectors \( u_{1} = (1, 1) \) and \( u_{2} = (2, 3) \), any vector \( v \) in this space can be written as \( v = c_{1}u_{1} + c_{2}u_{2} \), where \( c_{1} \) and \( c_{2} \) are your scalar coefficients.

In the exercise, for instance, we're trying to represent \( v = (4, -3) \) as a linear combination of \( u_{1} \) and \( u_{2} \). By solving for \( c_{1} \) and \( c_{2} \), we translate \( v \) into terms of the basis vectors, effectively finding its position in the coordinate system defined by those base vectors. Understanding linear combinations helps in comprehending how different vectors relate to one another within a vector space.

Solving a linear system is about finding the values of variables that meet all the given conditions, represented as equations. Each equation in a system tells us something about the relationships between those variables. In the context of the original exercise, we had to solve two equations to find the coefficients \( c_{1} \) and \( c_{2} \):

• \( c_{1} + 2c_{2} = 4 \)
• \( c_{1} + 3c_{2} = -3 \)

From these, you find \( c_{2} \), and use it to determine \( c_{1} \).

This process of solving linear systems is fundamental in fields like computer graphics, physics, and engineering where interpreting relationships between variable elements is crucial. Essentially, it's about finding the point where various lines or planes, represented by equations, intersect in the space, which provides us meaningful, real-world solutions.