Question

Consider the vector functions \(\phi_{1}\) and \(\phi_{2}\) defined by $$ \phi_{1}(t)=\left(\begin{array}{l} t \\ 1 \end{array}\right) \text { and } \phi_{2}(t)=\left(\begin{array}{c} t e^{t} \\ e^{t} \end{array}\right) $$ respectively. Show that the constant vectors \(\phi_{1}\left(t_{0}\right)\) and \(\phi_{2}\left(t_{0}\right)\) are linearly dependent for each \(t_{0}\) in the interval \(0 \leq t \leq 1\), but that the vector functions \(\phi_{1}\) and \(\phi_{2}\) are linearly independent on \(0 \leq t \leq 1 .\)

Short answer

In summary, the constant vectors \(\phi_{1}(t_{0})\) and \(\phi_{2}(t_{0})\) are linearly dependent for each \(t_{0}\) in the interval \(0 \leq t{\leq}1\), as there exists a non-trivial linear combination that results in the zero vector. However, the vector functions \(\phi_{1}(t)\) and \(\phi_{2}(t)\) are linearly independent on the same interval, as the only solution for the system of equations is the trivial one, \(c_{1}=0\) and \(c_{2}=0\).

Step-by-step solution

Determine constant vectors at \(t_{0}\)

First, we need to find the constant vectors \(\phi_{1}(t_{0})\) and \(\phi_{2}(t_{0})\) at any \(t_{0}\) in the interval \(0 \leq t \leq 1\). By substituting \(t=t_{0}\) in the given vector functions, we get: $$ \phi_{1}(t_{0})=\left(\begin{array}{l} t_{0} \\ 1 \end{array}\right) \text { and } \phi_{2}(t_{0})=\left(\begin{array}{c} t_{0} e^{t_{0}} \\ e^{t_{0}} \end{array}\right) $$

Check linear dependency for constant vectors

Now we need to check if there exists a non-trivial linear combination between the constant vectors at \(t_{0}\): Let's assume: $$ c_{1}\phi_{1}(t_{0})+c_{2}\phi_{2}(t_{0})=\mathbf{0} $$ Multiplying the coefficients and adding the vectors, we get: $$ \left(\begin{array}{l} c_{1}t_{0}+c_{2}t_{0} e^{t_{0}} \\ c_{1}+c_{2}e^{t_{0}} \end{array}\right)=\left(\begin{array}{l} 0 \\ 0 \end{array}\right) $$ From the above equation, we can find the coefficients: $$ \begin{cases} c_{1}t_{0}+c_{2}t_{0} e^{t_{0}}=0 \\ c_{1}+c_{2}e^{t_{0}}=0 \end{cases} $$ Dividing the first equation by \(t_{0}\) and using the second equation to solve for \(c_{1}\), we get: $$ c_{1}= -c_{2}e^{t_{0}} $$ Substituting this value back in the equation, we have a non-trivial solution (\(c_{1} \neq 0\) and \(c_{2} \neq 0\)) when: $$ -c_{2} e^{t_{0}}+c_{2}t_{0} e^{t_{0}}=0 \\ c_{2} e^{t_{0}} (t_{0}-1) = 0 \\ c_{2} = 0 \text{ or } t_{0} = 1 $$ Since there exists a non-trivial linear combination that results in the zero vector, the constant vectors \(\phi_{1}(t_{0})\) and \(\phi_{2}(t_{0})\) are linearly dependent for each \(t_{0}\) in the interval \(0 \leq t \leq 1\).

Check linear independence for vector functions

Now, we need to check whether \(\phi_1(t)\) and \(\phi_2(t)\) are linearly independent by verifying if the only linear combination that results in the zero vector function is the trivial combination: Let's assume: $$ c_{1}\phi_{1}(t)+c_{2}\phi_{2}(t)=\mathbf{0} $$ Multiplying the coefficients and adding the functions, we get: $$ \left(\begin{array}{l} c_{1}t+c_{2}t e^{t} \\ c_{1}+c_{2}e^{t} \end{array}\right)=\left(\begin{array}{l} 0 \\ 0 \end{array}\right) $$ From this equation, we have a system of equations: $$ \begin{cases} c_{1}t+c_{2}t e^{t}=0 \\ c_{1}+c_{2}e^{t}=0 \end{cases} $$ The first equation gives: $$ c_{1}+c_{2}e^{t} = 0 $$ Based on the second equation, we have \(c_{1} = -c_{2}e^{t}\). As \(c_{1}\) depends on the variable \(t\), this means that the only solution for this system of equations is the trivial one, \(c_{1}=0\) and \(c_{2}=0\). Therefore, the vector functions \(\phi_{1}\) and \(\phi_{2}\) are linearly independent on the interval \(0 \leq t \leq 1\).

Key concepts

Vector Functions

A vector function is essentially a function which takes variables, often called parameters, and maps them to a set of vectors. In our exercise, the vector functions \( \phi_1(t) \) and \( \phi_2(t) \) take the parameter \( t \) and return vectors. These vectors change as \( t \) changes, meaning they both describe curves or paths in a vector space.

Key points to understand about vector functions include:

• They are defined over a domain, which in this case is \( 0 \leq t \leq 1 \).
• They result in an output vector for every point in their domain.
• They can represent lines, curves, or surfaces depending on their formulation.

Understanding vector functions is crucial in solving differential equations and performing vector analysis as these functions describe the behavior of systems over time.

Linear Dependency

Linear dependency is a fundamental concept in linear algebra. It answers the question: Can a vector be composed as a linear combination of others? In the context of the constant vectors \( \phi_1(t_0) \) and \( \phi_2(t_0) \), they are considered linearly dependent if there are non-zero constants, \( c_1 \) and \( c_2 \), such that:
\[ c_1 \phi_1(t_0) + c_2 \phi_2(t_0) = \mathbf{0} \]

This means that one vector is essentially a scaled version of the other or a combination of others, and in this exercise, it's illustrated that the vectors \( \phi_1(t_0) \) and \( \phi_2(t_0) \) are linearly dependent for each \( t_0 \) due to the existence of such constants. It's important to note that linear dependency depends on the specific instance of \( t_0 \), and it's distinct from the independence of the vector functions themselves over the interval.

Differential Equations

Differential equations involve derivative expressions and are essential for modeling situations where change is continuous and time-sensitive. In our case, understanding the independence of vector functions \( \phi_1(t) \) and \( \phi_2(t) \) on an interval is useful.

Linear independence in this context means that the only solution to
\[ c_1 \phi_1(t) + c_2 \phi_2(t) = \mathbf{0} \]
for all \( t \) is the trivial solution \( c_1 = 0 \) and \( c_2 = 0 \). This indicates the vector functions do not form a basis that is redundant for describing any set of solutions of differential equations over their domain. Thus, they can each be utilized to provide unique insight or solutions when solving differential equations.

Vectors Analysis

Vector analysis is a critical tool in the realm of mathematics and physics, offering a range of techniques to work with scalar and vector fields. By determining the linear dependency or independence of vector functions, such as \( \phi_1(t) \) and \( \phi_2(t) \), we can conclude whether these vectors span a particular space or not.

The significance of the analysis boils down to these points:

• Independent vector functions can be used to form the basis for the vector space in question.
• These analyses support complex applications, including fluid dynamics, electromagnetism, and more.
• It's a precursor to solving various complex problems in applied mathematics and engineering.

Through vector analysis, determining the nature of these vector functions helps define how systems evolve and interact across the defined domain.