Problem 9 - Differential Equations and Linear Algebra

Question

Show that the given vector functions are linearly dependent on \((-\infty, \infty)\). $$\begin{aligned} &\mathbf{x}_{1}(t)=\left[\begin{array}{c} \sin ^{2} t \\ \cos ^{2} t \\ 2 \end{array}\right], \quad \mathbf{x}_{2}(t)=\left[\begin{array}{c} 2 \cos ^{2} t \\ 2 \sin ^{2} t \\ 1 \end{array}\right]\\\ &\mathbf{x}_{3}(t)=\left[\begin{array}{l} 2 \\ 2 \\ 5 \end{array}\right] \end{aligned}$$

Step-by-Step Solution

Answer

We found non-trivial constants \(c_1 = -4, c_2 = 3, c_3 = 1\) that satisfy the linear combination equation: \(-4\mathbf{x}_1(t) + 3\mathbf{x}_2(t) + 1\mathbf{x}_3(t) = \mathbf{0}\) Thus, the given vector functions are linearly dependent on the interval \((-\infty , \infty)\).

Step 1: Set up the linear combination

The linear combination of the vector functions can be written as: $$c_1\mathbf{x}_1(t) + c_2\mathbf{x}_2(t) + c_3\mathbf{x}_3(t) = \mathbf{0}$$

Step 2: Write the corresponding system of equations

From the linear combination equation, we get the following system of equations for each of the components of the vectors: 1. \(c_1\sin^2{t} + 2c_2\cos^2{t} + 2c_3 = 0\) 2. \(c_1\cos^2{t} + 2c_2\sin^2{t} + 2c_3 = 0\) 3. \(2c_1 + c_2 + 5c_3 = 0\)

Step 3: Solve the system of equations

Let's focus on the first two equations. Notice that adding them results in: \(c_1 + 2c_2 + 2c_1 + 2c_2 = 0\) Simplifying the equation: \(3c_1 + 4c_2 = 0\) We observe that the third equation is identical to the sum of the first two equations: \(2c_1 + c_2 + 5c_3 = 3c_1 + 4c_2 = 0\) So, any values of \(c_1\) and \(c_2\) that satisfy \(3c_1 + 4c_2 = 0\) will also satisfy the third equation, which means that the system has infinitely many solutions. Let's choose \(c_2 = 3\), then the equation becomes: \(3c_1 + 4(3) = 0\) Solving for \(c_1\): \(c_1 = -4\) Now we substitute \(c_1\), and \(c_2\) back into the third equation and solve for \(c_3\): \(2(-4) + 3 + 5c_3 = 0\) \(c_3 = 1\)

Step 4: Verify the constants are non-trivial

We have found the non-trivial constants \(c_1 = -4, c_2 = 3, c_3 = 1\), which satisfy the linear combination equation: \(-4\mathbf{x}_1(t) + 3\mathbf{x}_2(t) + 1\mathbf{x}_3(t) = \mathbf{0}\) Since the constants are not all zero, we can conclude that the given vector functions are linearly dependent on the interval \((-\infty , \infty)\).

Key Concepts

Linear Combination

A linear combination in linear algebra involves an equation where vectors are multiplied by scalar values and then added together. For instance, consider vectors \textbf{a}, \textbf{b}, and \textbf{c} in a vector space. A linear combination of these vectors can be expressed as
\(k_1\textbf{a} + k_2\textbf{b} + k_3\textbf{c}\),
where \(k_1\), \(k_2\), and \(k_3\) are scalars (real numbers). In the context of the exercise, we looked at a linear combination of vector functions to determine linear dependence. When such a combination equates to the zero vector and not all scalars are zero, it implies that the vector functions are linearly dependent. This concept is crucial for understanding the relationships between vectors in a vector space.

System of Equations

A system of equations is a set of multiple equations, each with the same set of variables. Our goal is to find a set of values for these variables that will satisfy all the given equations simultaneously. In the context of vector functions and their linear dependence, we equated a linear combination of the functions to a zero vector, which resulted in a system of equations. Each component in the vectors gave us one equation, allowing us to investigate the coefficients (scalars) that could combine the vector functions to reach the zero vector, and thus determine the linear dependence of the vector functions.

Solving Linear Systems

Solving a system of linear equations means finding the values for the variables that make all the equations true simultaneously. There are various methods to solve such systems, including substitution, elimination, and matrix operations like row reduction. In our solution, we used elimination by adding certain equations to eliminate variables, taking advantage of the trigonometric identities \(\sin^2{t} + \cos^2{t} = 1\). Solving the system helped us find specific scalars that, when used in the linear combination of provided vector functions, confirmed their linear dependence. This step is foundational to understanding how different vector functions can be related within the scope of a vector space.

Vector Space

A vector space, also known as a linear space, is a collection of vectors where two operations, vector addition and scalar multiplication, are defined and obey specific rules. These rules, or axioms, include the properties of associativity, commutativity, distributivity, and the existence of an additive identity and inverses. Within a vector space, we can discuss concepts like linear combination, linear independence, subspaces, bases, and dimension. Understanding whether given vector functions are linearly dependent or independent helps us determine the structure of the vector space, as well as solve various mathematical and real-world problems.