Problem 21 - Advanced Engineering Mathematics

Question

Determine whether the given set of functions is linearly dependent or linearly independent on the interval \((-\infty, \infty)\). $$ f_{1}(x)=1+x, \quad f_{2}(x)=x, \quad f_{3}(x)=x^{2} $$

Step-by-Step Solution

Answer

The functions are linearly independent on \((- fty, \infty)\).

Step 1: Define Linear Dependence

A set of functions \( \{ f_1, f_2, f_3 \} \) is linearly dependent if there exist constants \( c_1, c_2, c_3 \), not all zero, such that \( c_1 f_1(x) + c_2 f_2(x) + c_3 f_3(x) = 0 \) for all \( x \). Otherwise, they are linearly independent.

Step 2: Set Up the Linear Combination Equation

Consider the linear combination \( c_1 (1+x) + c_2 x + c_3 x^2 = 0 \). This implies \( c_1 + c_1 x + c_2 x + c_3 x^2 = 0 \). Rearrange it as \( c_1 + (c_1 + c_2)x + c_3 x^2 = 0 \).

Step 3: Compare Coefficients

For the equation \( c_1 + (c_1 + c_2)x + c_3 x^2 = 0 \) to hold for all \( x \), the coefficients of \( x^0, x^1, \) and \( x^2 \) must be zero. Thus, we get the system of equations: \( c_1 = 0 \), \( c_1 + c_2 = 0 \), and \( c_3 = 0 \).

Step 4: Solve the System of Equations

Solve the system: \( c_1 = 0 \), \( c_1 + c_2 = 0 \) implies \( c_2 = 0 \) due to \( c_1 = 0 \). The third equation \( c_3 = 0 \). Thus, \( c_1 = c_2 = c_3 = 0 \).

Step 5: Conclude Linear Independence

Since the only solution to the system of equations is \( c_1 = c_2 = c_3 = 0 \), the functions are linearly independent, as no non-trivial linear combination gives the zero function.

Key Concepts

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations. It's the backbone of much of modern mathematics and has applications in various fields such as computer science, physics, and engineering.
At its core, linear algebra is concerned with linear relations. These are equations where each term is either a constant or the product of a constant and a variable. A pivotal concept in linear algebra is linear independence, which tells us whether a set of vectors (or functions, in some contexts) are independent.
If a set of vectors is linearly independent, it means that no vector in the set can be written as a combination of the others. This concept is crucial in understanding the structure of vector spaces and is used to determine the dimension of the space. In practical terms, determining linear independence helps us understand if we have enough 'directions' to cover a vector space.

System of Equations

A system of equations is a set of equations with multiple variables. These equations are meant to be solved simultaneously. The solution to a system of equations is a set of values for the variables that satisfies all the equations in the system.
In our exercise, step 3 leads us to set up a system of equations by comparing coefficients from each term in the polynomial equation.

• We set \( c_1 = 0 \) for constant terms.
• The coefficient of \( x \) gives \( c_1 + c_2 = 0 \).
• Finally, \( c_3 = 0 \) for the quadratic term.

Solving these equations collectively allows us to determine the values of parameters that make the polynomial expression zero for every \( x \). For this exercise, finding that all parameters \( c_1, c_2, \,\, \text{and} \,\, c_3 \) are zero indicates that the functions are linearly independent.

Polynomial Functions

Polynomial functions are expressions involving a sum of powers of one or more variables multiplied by coefficients. The degree of a polynomial function is the highest power of the variable. In our example, we are dealing with polynomial functions of varying degrees for analysis.

• \( f_{1}(x) = 1+x \): This is a linear polynomial as it has the degree of 1.
• \( f_{2}(x) = x \): Another linear polynomial, representing the simplest form of a polynomial - a line passing through the origin.
• \( f_{3}(x) = x^2 \): This is a quadratic polynomial, with a degree of 2, presenting a parabola when plotted.

In the context of checking for linear independence, we look at polynomial functions to see if their linear combinations form a zero function only in a trivial case, i.e., when all coefficients are zero. This step highlights the importance of polynomial degrees and coefficients in deciding a function set's independence.