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Chapter 3: Problem 2

Find the coordinate vector of the given vector relative to the indicated ordered basis. \([-2,4]\) in \(\mathbb{R}^{2}\) relative to \(\left([0,-2],\left[-\frac{1}{2}, 0\right]\right)\)

Short Answer

Expert verified
The coordinate vector is \([-2, 4]\).

Step by step solution

01

Understand the Problem

You need to find the coordinate vector of \([-2, 4]\) in \(\mathbb{R}^2\) relative to the basis \(([0, -2], [-\frac{1}{2}, 0])\). This means expressing \([-2, 4]\) as a linear combination of the given basis vectors.
02

Set Up the Equation

Express \([-2, 4]\) as a linear combination \[c_1 \cdot [0, -2] + c_2 \cdot \left[-\frac{1}{2}, 0\right] = [-2, 4].\] Your goal is to find the coefficients \(c_1\) and \(c_2\) that satisfy this equation.
03

Solve for First Component

Focus on the first component of the vectors: \(-2 = c_1 \cdot 0 + c_2 \cdot \left(-\frac{1}{2}\right).\) Simplifying, this gives \[c_2 = 4.\]
04

Solve for Second Component

Now consider the second component: \4 = c_1 \cdot (-2) + c_2 \cdot 0.\ Substituting \(c_2 = 4\) from the previous step and simplifying, you get \[-2c_1 = 4.\] Solving for \(c_1\), you find \[c_1 = -2.\]
05

Write the Final Coordinate Vector

Conclude that the coordinate vector of \([-2, 4]\) relative to the given basis is \([-2, 4]\). This means that \([-2, 4]\) can be expressed as \(-2 \times [0, -2] + 4 \times \left[-\frac{1}{2}, 0\right].\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Combination
In the world of vector spaces, linear combination is a foundational concept. When we say a vector is a linear combination of other vectors, we mean it can be expressed as a sum where each of the other vectors is multiplied by a specific coefficient. For the exercise, the vector \([-2, 4]\) is expressed in terms of basis vectors \([0, -2]\) and \([-\frac{1}{2}, 0]\). Here is how it works:
  • The vector \([-2, 4]\) is expressed as \(c_1 \cdot [0, -2] + c_2 \cdot \left[-\frac{1}{2}, 0\right]\).
  • Your task is to identify the coefficients \(c_1\) and \(c_2\) such that this equation holds true.
  • These coefficients, \(c_1 = -2\) and \(c_2 = 4\), effectively create the correct linear combination.
Finding these coefficients helps us express one vector uniquely in the context of given basis vectors.
Basis
A basis in a vector space gives us a way to describe every vector in that space. It is a set of vectors that are linearly independent and span the vector space. Let's break this down:
  • **Linearly Independent**: No vector in the set can be expressed as a linear combination of the others. This feature ensures that each basis vector adds a unique dimension.
  • **Span the Vector Space**: Any vector in the space can be expressed as a linear combination of the basis vectors.
In this exercise:
  • Our basis consisted of two vectors \([0, -2]\) and \([-\frac{1}{2}, 0]\).
  • These vectors are linearly independent, meaning neither can be formed from the other by multiplication of a scalar.
  • These basis vectors together span \(\mathbb{R}^2\), allowing any vector within \(\mathbb{R}^2\) to be expressed using a unique linear combination of this basis.
Vector Spaces
A vector space is a broader context in which vectors exist, and it consists of vectors that can be added together and multiplied by scalars. Key properties of vector spaces include:
  • Vectors can be added together (Closure under addition).
  • Vectors can be multiplied by numbers (Closure under scalar multiplication).
  • It includes a zero vector, which acts like zero in addition.
The exercise deals with vector spaces in two dimensions, specifically \(\mathbb{R}^2\):
  • In \(\mathbb{R}^2\), vectors have two components (like \([-2, 4]\)).
  • Vectors can interact under addition and scalar multiplication to form new vectors in the same space.
Understanding vector spaces helps explore concepts like span, basis, and dimensions, maintaining the structure and features of the space.
Mathematics Education
Understanding vectors and their properties is a critical area in mathematics education. When students learn about vectors, they develop analytical skills crucial for problem-solving. Here's why focusing on vectors is essential in education:
  • **Abstract Thinking**: Understanding vector spaces helps students think abstractly and deal with complex mathematical concepts.
  • **Real-World Applications**: Vectors apply to various real-world situations, including physics, engineering, and computer science.
  • **Critical Thinking**: Solving vector-related problems enhances students' critical thinking abilities, as they must analyze relationships and formulate strategies to find solutions.
  • **Foundation for Advanced Topics**: Mastering basic vector concepts lays the groundwork for exploring more advanced topics in mathematics such as linear algebra and calculus.
Integrating vector education into learning helps students build a strong mathematical foundation, preparing them for academic success and practical applications.

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Most popular questions from this chapter

Let \(V\) be an inner-product space, and let \(S\) be a subset of \(V\). Prove that $$ S^{L}= $$ \(\\{v \in V \mid v\) is orthogona! to each vector in \(S\\}\) is a subspace of \(V\).

In Exercises 22-24, find a basis for the given subspace of the vector space. \(\operatorname{sp}\left(1, \sin ^{2} x, \cos 2 x, \cos ^{2} x\right)\) in \(F\)

Let \(u\) and \(v\) be vectors in an inner-product space, and suppose that \(\|\mathrm{u}\|=3\) and \(\|v\|=5\). Find \(\langle u+2 v, u-2 v\rangle\).

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Mark each of the foliowing True or Faise. ___a. Matrix multiplication is a vector-space operation on the set \(M_{m \times n}\) of all \(m \times n\) matrices. ___b. Matrix multiplication is a vector-space operation on the set \(M_{n}\) of all square \(n \times n\) matrices. ___c. Multiplication of any vector by the zero scalar always yields the zero vector. ___d. Multiplication of a nonzero vector by a nonzero scalar never yields the zero vector. ___e. No vector is its own additive inverse.f. The zero vector is the only vector that is its own additive inverse. ___g. Multiplication of two scalars is of no concern in the definition of a vector space. ___h. One of the axioms for a vector space relates addition of scalars, multiplication of a vector by scalars, and addition of vectors. ___i. Every vector space has at least two vectors. ___j. Every vector space has at least one vector.
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