Problem 19 - Advanced Engineering Mathematics

Question

Suppose we wish to determine whether the set of column vectors $$ \begin{array}{r} \mathbf{v}_{1}=\left(\begin{array}{l} 4 \\ 3 \\ 2 \\ 1 \end{array}\right), \quad \mathbf{v}_{2}=\left(\begin{array}{l} 1 \\ 2 \\ 2 \\ 1 \end{array}\right), \quad \mathbf{v}_{3}=\left(\begin{array}{r} -1 \\ 1 \\ 1 \\ 1 \end{array}\right) \\ \mathbf{v}_{4}=\left(\begin{array}{l} 2 \\ 3 \\ 4 \\ 1 \end{array}\right), \quad \mathbf{v}_{5}=\left(\begin{array}{r} 1 \\ 7 \\ -5 \\ 1 \end{array}\right) \end{array} $$ is linearly dependent or linearly independent. By Definition \(7.6 .3\), if $$ c_{1} \mathbf{v}_{1}+c_{2} \mathbf{v}_{2}+c_{3} \mathbf{v}_{3}+c_{4} \mathbf{v}_{4}+c_{5} \mathbf{v}_{5}=\mathbf{0} $$ only for \(c_{1}=0, c_{2}=0, c_{3}=0, c_{4}=0, c_{5}=0\), then the set of vectors is linearly independent; otherwise the set is linearly dependent. But (4) is equivalent to the linear system $$ \begin{array}{r} 4 c_{1}+c_{2}-c_{3}+2 c_{4}+c_{5}=0 \\ 3 c_{1}+2 c_{2}+c_{3}+3 c_{4}+7 c_{5}=0 \\ 2 c_{1}+2 c_{2}+c_{3}+4 c_{4}-5 c_{5}=0 \\ c_{1}+c_{2}+c_{3}+c_{4}+c_{5}=0 \end{array} $$ Without doing any further work, explain why we can now conclude that the set of vectors is linearly dependent.

Step-by-Step Solution

Answer

The set is linearly dependent because we have 5 vectors in 4-dimensional space.

Step 1: Review Definition of Linear Dependence

A set of vectors is linearly dependent if there exists a nontrivial combination of the vectors that results in the zero vector. This means not all of the coefficients \( c_i \) need to be zero.

Step 2: Assess the Number of Vectors vs Dimensions

We have 5 vectors \( \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4, \mathbf{v}_5 \) each with 4 components. According to the theory of linear algebra, a set of more than \( n \) vectors in \( n \) dimensions is always linearly dependent.

Step 3: Establishing Dependence by Dimensions

In a 4-dimensional space (each vector has four entries), we can have at most 4 linearly independent vectors. Since we have 5 vectors, this set must depend on each other, hence it is linearly dependent.

Step 4: Conclusion using the Linear System

The given linear system, if solved, would show that not all \( c_i \) must be zero because the number of vectors (5) exceeds the dimensions (4), indicating a dependency exists without needing actual calculation.

Key Concepts

Vector Spaces

A vector space is a fundamental concept in linear algebra, it consists of vectors that can be added together and multiplied by scalars. Vectors in a vector space follow specific rules of operation, such as associativity, commutativity of addition, and compatibility of scalar multiplication.

In our given exercise, we deal with column vectors \( \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4, \mathbf{v}_5 \) within a vector space. Here, the addition and scaling of these vectors abide by the rules of the vector space framework. This guarantees standardized operations irrespective of the representation of vectors.

Understanding vector spaces is crucial because they provide the environment where vectors live and interact, essentially forming the foundation for concepts like linear dependence and independence as seen in our problem.

Linear Algebra

Linear algebra is the branch of mathematics concerning vector spaces and the linear mappings between them. It focuses on concepts like systems of linear equations, matrices, determinants, and vector spaces.

In the context of this exercise, linear algebra provides the tools to analyze and characterize the set of vectors \( \{ \mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4, \mathbf{v}_5 \} \). By applying linear algebra techniques, we assess the relationships between vectors, such as determining if they form a linearly independent or dependent set.

• Linear Equations: Here, the system of equations derived from the vector set will help identify dependencies quickly.
• Matrix Representation: Vector sets can be transformed into a matrix form that organizes computation and eases the identification of dependencies.

Linear algebra is essential because it enables the structured and systematic approach to probing the behavior and properties of vector collections.

Dimension Theory

Dimension theory is about understanding the size or capacity of a vector space. Specifically, it's about the maximum number of linearly independent vectors in a space, which is known as its dimension.

In our exercise, each vector has four entries, hinting that they inhabit a four-dimensional space. According to the dimension theory, in a space of dimension \( n \), we can have at most \( n \) linearly independent vectors. This insight allows us to conclude quickly that a set of 5 vectors in a 4-dimensional space is surely dependent. An extra vector implies some redundancy or overlap, establishing a dependency.

The beauty of dimension theory lies in its ability to simplify complex vector relationships, aiding in the quick assessment of vector dependencies without delving into solving linear systems extensively.

Linear Independence

Linear independence is a concept used to determine if vectors in a set are fundamentally necessary for spanning a space.

Vectors are said to be linearly independent if the only way they sum to the zero vector is by each vector being multiplied by zero. If even one vector can be written as a combination of others, the set is dependent.

In our case, the solution signifies that the set of vectors is dependent. This is because five vectors in a four-dimensional space imply linear dependence - by definition, you cannot have more independent directions than the space's dimension allows. The linear equations formed by the vectors affirm this dependency without needing full computation.

• An understanding of linear independence helps pinpoint redundancy, ensuring maximum efficiency in vector representation.

Recognizing redundancy and optimizing vector sets are crucial skills in both theoretical studies and practical applications of vector spaces.