Question

A sum of scalar multiples of two or more vectors (such as \(c_{1} \mathbf{u}+c_{2} \mathbf{v}+c_{3} \mathbf{w},\) where \(c_{i}\) are scalars) is called a linear combination of the vectors. Let \(\mathbf{i}=\langle 1,0\rangle, \mathbf{j}=\langle 0,1\rangle\) \(\mathbf{u}=\langle 1,1\rangle,\) and \(\mathbf{v}=\langle-1,1\rangle\) Express \langle 4,-8\rangle as a linear combination of \(\mathbf{u}\) and \(\mathbf{v}\).

Short answer

Question: Express the vector \(\langle 4, -8\rangle\) as a linear combination of the vectors \(\mathbf{u}=\langle 1,1\rangle\) and \(\mathbf{v}=\langle-1,1\rangle\). Answer: \(\langle 4, -8\rangle = -2\mathbf{u} - 6\mathbf{v}\)

Step-by-step solution

Write down the given vectors and the equation to solve

Given vectors: \(\mathbf{u}=\langle 1,1\rangle\) \(\mathbf{v}=\langle-1,1\rangle\) Target vector: \(\langle 4,-8\rangle\) We want to find \(c_1\) and \(c_2\) such that: \langle 4,-8\rangle = \(c_1 \mathbf{u} + c_2 \mathbf{v}\)

Set up the system of linear equations

Using the equation from Step 1, we can set up the following system of linear equations: \(c_1(1) + c_2(-1) = 4\) \(c_1(1) + c_2(1) = -8\)

Solve the system of linear equations

We can solve the two linear equations for \(c_1\) and \(c_2\): 1. Add equation 1 and equation 2: \(c_1 - c_2 + c_1 + c_2 = 4 - 8\) \(2c_1 = -4\) Divide both sides of the equation by 2: \(c_1 = -2\) 2. Substitute \(c_1\) back into the first equation: \(-2 - c_2 = 4\) Add 2 to both sides of the equation: \(-c_2 = 6\) Multiply both sides of the equation by -1: \(c_2 = -6\)

Write the linear combination

Now that we have the values for \(c_1\) and \(c_2\), we can write the linear combination of \(\mathbf{u}\) and \(\mathbf{v}\) that equals \langle 4, -8\rangle: \langle 4,-8\rangle = \((-2) \mathbf{u} + (-6) \mathbf{v}\)

Key concepts

Vectors

Vectors are fundamental elements in linear algebra, representing quantities that have both direction and magnitude. Imagine arrows pointing in specific directions; these are vectors. In most mathematical formulations, vectors are expressed as ordered pairs or triples, such as \( \langle 1, 0 \rangle \). Here, the numbers represent the vector's components in a specific coordinate system. A vector like \( \mathbf{i} = \langle 1, 0 \rangle \) or \( \mathbf{j} = \langle 0, 1 \rangle \) is used to express movement along the axes in a plane.
When working with vectors in exercises like this one, it's common to be asked to combine or manipulate them in various ways. Understanding vectors enables us to describe physical quantities like velocity, force, and even growth rates in different contexts. They are not just abstract entities but also practical tools to model real-world phenomena in a structured form using mathematics.

Linear Combination

A linear combination involves creating a new vector by adding together scalar multiples of given vectors. Scalars are simply numbers that scale the vectors. For example, a linear combination of two vectors \( \mathbf{u} \) and \( \mathbf{v} \) can be written as \( c_1 \mathbf{u} + c_2 \mathbf{v} \), where \( c_1 \) and \( c_2 \) are scalars. The combination forms a new vector, which lies in the span of the original vectors.
In our example, we used the linear combination concept to express the vector \( \langle 4, -8 \rangle \) as the sum of multiples of vectors \( \mathbf{u} = \langle 1, 1 \rangle \) and \( \mathbf{v} = \langle -1, 1 \rangle \). We calculated the scalars \( c_1 = -2 \) and \( c_2 = -6 \) to achieve this.
The idea of linear combination is potent as it forms the basis for understanding vector spaces and linear transformations, foundational elements in linear algebra. This concept hints at the possibility of representing any vector in a vector space as a combination of a set of vectors, a topic developed further in more advanced studies.

Systems of Equations

Systems of equations are sets of equations that are solved together as a group. In linear algebra, these equations usually involve vectors and are a natural way to express problems involving linear combinations. Solving these systems means finding values, often called solutions, that satisfy all equations simultaneously.
In this problem, we formed a system of two equations to determine the scalars for the linear combination, which enabled us to express the vector \( \langle 4, -8 \rangle \) using vectors \( \mathbf{u} \) and \( \mathbf{v} \). The equations were:

• \( c_1(1) + c_2(-1) = 4 \)
• \( c_1(1) + c_2(1) = -8 \)

Solving this system involved basic algebraic operations, like adding and substituting, to find \( c_1 = -2 \) and \( c_2 = -6 \).
Understanding how to manipulate systems of equations is essential for tackling more complex linear algebra problems. This skill helps with finding intersections of lines or planes, and lies at the heart of problem-solving in fields ranging from engineering to economics.