Question

Suppose that the vectors \(\mathbf{x}^{(1)}, \ldots, \mathbf{x}^{(m)}\) each have \(n\) components, where \(n

Short answer

Answer: Yes, the given vectors are linearly dependent. This can be shown by setting up a homogeneous system of linear equations with the given vectors and finding a nontrivial solution. The matrix A, which is formed by taking the given vectors as columns, is an n x m matrix with n < m, and there will exist at least one nontrivial solution for this system. Therefore, the given vectors are linearly dependent.

Step-by-step solution

Create the homogeneous system of linear equations with the given vectors

To show that the given vectors \(\mathbf{x}^{(1)}, \ldots, \mathbf{x}^{(m)}\) are linearly dependent, we need to find scalars \(a_1, a_2, \ldots, a_m\) not all equal to zero, such that: \(a_1\mathbf{x}^{(1)} + a_2\mathbf{x}^{(2)} + \cdots + a_m\mathbf{x}^{(m)} = \mathbf{0}\) Where \(\mathbf{0}\) is the zero vector. To create a homogeneous system of linear equations, we can write this equation in matrix form by taking the given vectors as the columns of a matrix \(A\). \(A = [\mathbf{x}^{(1)}, \mathbf{x}^{(2)}, \ldots, \mathbf{x}^{(m)}]\) So the equation becomes, \(A \cdot \begin{bmatrix} a_1 \\ a_2 \\ \vdots \\ a_m \end{bmatrix} = \mathbf{0}\)

Analyze the homogeneous system and its solution set

We have set up a homogeneous system in the form \(A\mathbf{a} = \mathbf{0}\), where \(\mathbf{a}\) is the column vector with the coefficients \(a_1, a_2, \ldots, a_m\), and \(\mathbf{0}\) is the zero vector. Here, matrix \(A\) is an \(n \times m\) matrix with \(n < m\). Since there are more columns than rows in \(A\), by the pigeonhole principle, there is no unique solution to this homogeneous system. There must exist at least one nontrivial solution, meaning there are scalars \(a_1, a_2, \ldots, a_m\) not all equal to zero.

Conclude that the given vectors are linearly dependent

Since we have found a nontrivial solution to the homogeneous system \(A\mathbf{a} = \mathbf{0}\), this means there exists a nontrivial linear combination of \(\mathbf{x}^{(1)}, \mathbf{x}^{(2)}, \ldots, \mathbf{x}^{(m)}\) that results in the zero vector. Therefore, these vectors are linearly dependent.

Key concepts

Vectors

Understanding vectors is crucial when studying linear algebra or any form of mathematics that deals with quantities possessing both magnitude and direction. A vector in the context of real linear spaces can be imagined as an arrow. This arrow has certain values represented in ordered lists called components.
In our exercise, we have vectors \( \mathbf{x}^{(1)}, \ldots, \mathbf{x}^{(m)} \), each with \( n \) components. When you see \( \mathbf{x}^{(i)} \), think of it as an entity that can point in \( n \) dimensional space.

• A vector can be denoted by bold letters such as \( \mathbf{x} \).
• Components of a vector determine its position in multi-dimensional space.
• The length of the vector list called the dimension affects its properties.

Vectors play an important role in determining linear dependence, an essential concept reflecting how vectors are interconnected with each other through linear combinations.

Homogeneous System

A homogeneous system of equations is a system where each equation equates a linear combination of variables to zero. Why do we care about such systems? Because they help us understand relationships among vectors.
In the given exercise, to check if the vectors are linearly dependent, we set up the system \( a_1\mathbf{x}^{(1)} + a_2\mathbf{x}^{(2)} + \cdots + a_m\mathbf{x}^{(m)} = \mathbf{0} \).

• This system is called "homogeneous" because the right-hand side is always a zero vector.
• Homogeneous systems always have at least one solution, called the trivial solution, where all coefficients are zero.
• If there are more solutions than the trivial, the vectors are linearly dependent.

Analyzing homogeneous systems reveals not only the solution but also the nature of dependency between vectors.

Matrix Form

Representing vectors in matrix form is an efficient way to manage and analyze multiple vectors simultaneously. In this approach, each vector is treated as a column in a matrix.
In our system, we assemble vectors \( \mathbf{x}^{(1)}, \mathbf{x}^{(2)}, \ldots, \mathbf{x}^{(m)} \) into matrix \( A \) as its columns. This gives us an \( n \times m \) matrix, wherein \( n < m \).

• Matrix form enables straightforward calculation with vectors in linear equations.
• Each column vector in matrix \( A \) represents one of the vectors.
• The equation \( A\mathbf{a} = \mathbf{0} \) highlights the relationship between matrix \( A \) and vector \( \mathbf{a} \) (coefficients).

Using this form simplifies complex vector operations, making it easier to check for linear dependencies.

Nontrivial Solution

A nontrivial solution stands for a solution to a system of equations where not all variable coefficients are zero. This contrasts with the trivial solution where all these coefficients are zero.
In detailed terms, if you have the homogeneous system \( A\mathbf{a} = \mathbf{0} \) and find any solution besides the trivial one, then a nontrivial solution exists. Such a solution indicates that not all coefficients \( a_1, a_2, \ldots, a_m \) are zero.

• Finding a nontrivial solution confirms linear dependence among vectors.
• With \( n < m \), the Pigeonhole Principle implies that the system must have permutations leading to this nontrivial solution.
• It shows that a weighted sum of vectors can produce zero, meaning overlap in directional influence.

Recognizing when nontrivial solutions exist is a sign of vectors being linked, highlighting their dependence within the system.