Question

Determine whether the given set of vectors is linearly independent. If linearly dependent, find a linear relation among them. The vectors are written as row vectors to save space, but may be considered as column vectors; that is, the transposes of the given vectors may be used instead of the vectors themselves. $$ \mathbf{x}^{(1)}=(1,1,0), \quad \mathbf{x}^{(2)}=(0,1,1), \quad \mathbf{x}^{(3)}=(1,0,1) $$

Short answer

Answer: The given set of vectors is linearly independent.

Step-by-step solution

Set up the system of linear equations from the vectors

First, we will write a matrix from these vectors as columns, and then find its determinant. If the determinant is non-zero, the given vectors are linearly independent; otherwise, they are linearly dependent. Form the matrix \(A\) of the given vectors as columns: $$ A = \begin{pmatrix} 1 & 0 & 1 \\ 1 & 1 & 0 \\ 0 & 1 & 1 \\ \end{pmatrix} $$

Calculate the determinant for matrix A

Next, we will find the determinant of matrix A. The determinant is given by: $$ \det(A) = \begin{vmatrix} 1 & 0 & 1 \\ 1 & 1 & 0 \\ 0 & 1 & 1 \\ \end{vmatrix} = 1 \cdot \begin{vmatrix} 1 & 0 \\ 1 & 1 \\ \end{vmatrix} - 0 \cdot \begin{vmatrix} 1 & 0 \\ 0 & 1 \\ \end{vmatrix} + 1 \cdot \begin{vmatrix} 1 & 1 \\ 0 & 1 \\ \end{vmatrix} $$

Evaluate the determinant

Now, calculate the values inside the vertical bars: $$ \det(A) = 1 \cdot (1\cdot1 - 1\cdot0) - 0 \cdot (1\cdot1 - 0\cdot1) + 1 \cdot (1\cdot1 - 0\cdot1) \\ \det(A) = 1\cdot1 - 0\cdot1 + 1\cdot1 \\ \det(A) = 2 $$ Since the determinant of matrix A is non-zero (\(\det(A) = 2 \ne 0\)), it indicates that the given set of vectors is linearly independent. So the given set of vectors, \(\mathbf{x}^{(1)}, \mathbf{x}^{(2)},\) and \(\mathbf{x}^{(3)}\) are linearly independent.

Key concepts

Vectors

Vectors are fundamental objects in linear algebra, representing quantities that have both magnitude and direction. For instance, the vectors presented in the exercise, \(\mathbf{x}^{(1)}=(1,1,0)\), \(\mathbf{x}^{(2)}=(0,1,1)\), and \(\mathbf{x}^{(3)}=(1,0,1)\), can be visualized as arrows pointing in three-dimensional space.

Understanding vectors is crucial to solve myriad problems in physics, engineering, and mathematics. In the context of the exercise, determining whether a set of vectors is linearly independent involves finding if these vectors can be expressed as a combination of each other or if they stand unique in their dimensional space. As learners, it's essential to visualize these vectors and understand their interactions geometrically.

Determinant

The determinant is a value associated with square matrices, which gives us powerful insights into the properties of the matrix. It can tell us, for instance, if a matrix is invertible or if its column or row vectors are linearly independent.
In the solution provided, the determinant of matrix \(A\) was calculated to verify if the given vectors are linearly independent. A non-zero determinant, which in our example is \(2\), indicates that the vectors are unique and do not depend on each other. As a learner, it's crucial to understand that the determinant serves as a gateway to uncovering the matrix's underlying characteristics.

Linear Algebra

Linear algebra is the area of mathematics that deals with vectors, matrices, and linear transformations. It encompasses the understanding of linear systems, which are foundational in scientific and engineering disciplines.

When we say that vectors are linearly independent, like in our exercise, we mean that these vectors contribute to the richness of the space; they expand the space in which they exist. On the other hand, if they were linearly dependent, one vector could be written as a combination of another, offering nothing new to the space. Linear algebra techniques, such as calculating the determinant to assess linear independence, are essential tools for analyzing and interpreting multiple dimensions of data in real-world applications.