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Chapter 4: Problem 1

Describe the zero vector (the additive identity) of the vector space. $$R^{4}$$

Short Answer

Expert verified
The zero vector in \( R^{4} \) vector space is represented as (0, 0, 0, 0).

Step by step solution

01

Understanding the vector space

The given vector space is \( R^{4} \), which is the 4-dimensional real number space. Every vector in this space has four components.
02

Defining the zero vector

The zero vector in a vector space is the vector that, when added to any vector in the space, does not change the latter vector.
03

Representing the zero vector in \( R^{4} \)

The zero vector in \( R^{4} \) vector space can be represented as (0, 0, 0, 0). Each element in this 4-tuple represents a dimension (coordinate). When any vector in \( R^{4} \) adds with this zero vector, it doesn't change, hence satisfying the property of a zero vector.

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Key Concepts

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

Additive Identity
In mathematics, the concept of additive identity is key to understanding how numbers and vectors behave.
In any number set or vector space, there is a unique element known as the additive identity.
For numbers, this is 0, and for vectors, this is a special vector known as the zero vector. In a vector space, the zero vector is the additive identity.
  • It is denoted as a vector with all zero components.
  • When added to any other vector in the space, the result is that same vector.
This property is crucial because it means the zero vector acts like zero in regular arithmetic.
If you take any vector and add the zero vector to it, the vector remains unchanged.
This keeps the structure of vector spaces consistent and predictable, much like addition of numbers.
Vector Space
A vector space is a fundamental concept used throughout mathematics and science.
It provides the setting in which vectors can be added together and multiplied by numbers, or scalars. Every vector space must satisfy specific properties:
  • Vector addition: Adding two vectors in the space results in another vector in the same space.
  • Scalar multiplication: Multiplying a vector by a scalar results in another vector in the space.
  • Additive identity: There is a zero vector, acting as the identity element such that any vector plus this zero vector yields the original vector.
These properties ensure that vector spaces are closed under the operations of addition and scalar multiplication.
This means all operations performed within the space remain within that space, making calculations consistent and reliable.
R^4
When we talk about the vector space denoted by \(R^4\), we are referring to the four-dimensional real number space.
Each vector within \(R^4\) has exactly four components, and these components are real numbers.Understanding \(R^4\):
  • A vector in \(R^4\) might look like \((a, b, c, d)\), where \(a, b, c,\) and \(d\) are real numbers.
  • The zero vector in this space is \((0, 0, 0, 0)\), which acts as the additive identity.
In visualizing \(R^4\), things get tricky since we can only visually perceive up to three dimensions.However, it's helpful to think of each component in the vector as representing a specific dimension.
While it isn't possible to "see" four-dimensional vectors like three-dimensional ones, calculations are performed using the same principles as in lower dimensions, ensuring mathematical consistency.

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Most popular questions from this chapter

Perform a rotation of axes to eliminate the \(x y\) -term, and sketch the graph of the conic. $$x y+1=0$$

Find the Wronskian for the set of functions. $$\left\\{1, e^{x}, e^{2 x}\right\\}$$

Perform a rotation of axes to eliminate the \(x y\) -term, and sketch the graph of the "degenerate" conic. $$x^{2}-2 x y+5 y^{2}=0$$

Use a graphing utility or computer software program with matrix capabilities to find the transition matrix from \(B\) to \(B^{\prime}\) $$B=\\{(-2,1),(3,2)\\}, B^{\prime}=\\{(1,2),(-1,0)\\}$$

Use Theorem 4.21 to (a) find the transition matrix from \(B\) to \(B^{\prime},\) (b) find the transition matrix from \(B^{\prime}\) to \(B\) (c) verify that the two transition matrices are inverses of each other, and (d) find \([\mathbf{x}]_{a}\) when provided with \([\mathbf{x}]_{B^{*}}\) $$B=\\{(1,0,2),(0,1,3),(1,1,1)\\}$$ $$B^{\prime}=\\{(2,1,1),(1,0,0),(0,2,1)\\},$$ $$[\mathbf{x}]_{B^{\prime}}=\left[\begin{array}{r}1 \\ 2 \\\ -1\end{array}\right]$$
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