Question

In each of Problems 1 through 8 determine whether the given pair of functions is linearly independent or linearly dependent. \(f(t)=3 t, \quad g(t)=|t|\)

Short answer

The functions \(f(t)=3t\) and \(g(t)=|t|\) are linearly independent because they cannot be written as a nontrivial linear combination of one another. The only solution for the constants \(c_1\) and \(c_2\) in the equation \(c_1(3t) + c_2|t| = 0\) is the trivial solution where \(c_1=0\) and \(c_2=0\).

Step-by-step solution

Establish whether these functions can be written as a linear combination

To check whether \(f(t)\) can be written as a linear combination of \(g(t)\), we need to find constants \(c_1\) and \(c_2\) such that \(c_1f(t) + c_2g(t) = 0\). Assume such constants exist, then we have: $$c_1(3t) + c_2|t| = 0$$

Solve for constants \(c_1\) and \(c_2\)

Now, we have to examine two cases: \(t \geq 0\) and \(t < 0\). Case 1: \(t \geq 0\) In this case, \(|t|=t\). Then we have: $$c_1(3t) + c_2t = (3c_1 + c_2)t = 0$$ As the above equation must hold for all \(t \geq 0\), we have: $$3c_1 + c_2 = 0$$ Case 2: \(t < 0\) In this case, \(|t|=-t\). Then we have: $$c_1(3t) - c_2t = (3c_1 - c_2)t = 0$$ As the above equation must hold for all \(t < 0\), we have: $$3c_1 - c_2 = 0$$

Solve the system of equations for the constants \(c_1\) and \(c_2\)

Now we have a system of equations: $$3c_1 + c_2 = 0$$ $$3c_1 - c_2 = 0$$ This system has the unique solution \(c_1=\frac{c_2}{3} = 0\), and \(c_2=0\).

Determine if the functions are linearly independent or dependent

Since the only solution to the system of equations is the trivial solution \(c_1=0\) and \(c_2=0\), we can conclude that the functions \(f(t)=3t\) and \(g(t)=|t|\) are linearly independent.

Key concepts

Differential Equations

Differential equations are mathematical equations that involve functions and their derivatives. They play a crucial role in modeling various phenomena where rates of change are important. From physics to biology, differential equations help describe the relationships between changing quantities. By solving these equations, we can understand how a system evolves over time.

When we solve a differential equation, we aim to find a function or a set of functions that satisfy the given equation. These solutions often involve constants of integration, which are determined based on initial conditions or specific constraints. Thus, differential equations are not just about finding one solution, but a family of solutions that adhere to certain criteria.

Differential equations can range from very simple to highly complex. Linear differential equations are a simpler form, which allows for straightforward techniques of solution. They offer a way to understand many real-world systems, especially when approximations can be made to simplify non-linear systems into linear ones. Understanding the difference between linearly dependent and independent solutions in this context is essential.

Linearly Dependent

In mathematics, and particularly in linear algebra and vector calculus, the concept of linear dependence describes a situation where a set of functions can be expressed as a linear combination of other functions in the set using non-trivial coefficients.

For example, if you have functions like in the given exercise, if you can write one function as a combination of others using coefficients that are not all zero, then those functions are linearly dependent. Being linearly dependent implies redundancy in your set because one function doesn't bring any new information compared to the others.

• If functions are linearly dependent: It means one function is a multiple or a combination of others.
• If functions are linearly independent: Each function brings unique, indispensable information to the table.

To determine dependency, we form equations by combining functions, setting them equal to zero, and finding constants that make this equation true. If the only solution is when all coefficients are zero, the functions are independent. This is a critical check in mathematics as linearly independent sets form a basis for vector spaces.

Linear Combination

A linear combination involves forming new expressions by multiplying given functions by constant coefficients and adding the results. This mathematical concept is vital across various areas such as algebra, differential equations, and vector spaces.

The crux of solving problems involving linear combinations is finding the set of coefficients that can make your combination match a specific requirement, often achieving zero as the result, like in the problem of checking linear dependence.

• Linear combinations help in spanning spaces, showing the extent or capability of a set of vectors/functions.
• They are powerful in function approximation and transformation from one basis to another.

A typical scenario, as seen in the exercise, is to establish whether the functions can be formed using non-zero coefficients or if they naturally form a zero combination only with zero coefficients. The former implies dependence, while the latter signifies independence, helping in exploring the dimensionality and uniqueness brought by a set of functions.