Linearly independent functions cannot be expressed as a linear combination of each other. In the case of the two functions f(x) = x and g(x) = sin(x), they are linearly independent because there are no constants c₁ and c₂ (not both zero) that can make c₁ ⋅ f(x) + c₂ ⋅ g(x) = 0 for all x.
Graphically, this means that there is no way to combine these two functions linearly to produce the zero function (a horizontal line at y = 0). The functions are not redundant and cannot be derived from each other through linear combinations.
Linearly independent functions cannot be expressed as a linear combination of each other. In the case of the two functions f(x) = x and g(x) = sin(x), they are linearly independent because the ratio of f(x) to g(x) is not constant for all x.
If the functions were linearly dependent, the ratio f(x)/g(x) would be constant. However, as x varies, the value of sin(x) oscillates between -1 and 1, while x continues to increase or decrease linearly. Therefore, the ratio x/sin(x) does not remain constant, and the functions are linearly independent.
This ratio perspective emphasizes that linearly independent functions do not maintain a fixed proportional relationship across their entire domain, further illustrating that they cannot be derived from each other through linear combinations.
Definitions and Concepts
- Linear Independence: A set of functions is linearly independent if there are no constants c₁, c₂, c₃ (not all zero) that can make the equation c₁ ⋅ f₁(x) + c₂ ⋅ f₂(x) + c₃ ⋅ f₃(x) = 0 true for all x.
Example of Three Linearly Independent Functions
- f₁(x) = x
- f₂(x) = sin(x)
- f₃(x) = eˣ
Explanation
- Write the equation for linear dependence: c₁ ⋅ f₁(x) + c₂ ⋅ f₂(x) + c₃ ⋅ f₃(x) = 0.
- Try to find constants c₁, c₂, c₃ (not all zero) that make the equation true for all x.
- Observe that there is no way to choose the constants to make the equation true for all x. The functions are fundamentally different, and no linear combination of them will collapse to the zero function (a horizontal line at y = 0).
- Since there are no constants that satisfy the equation for all x, the functions are linearly independent.
Graphical Interpretation
If the functions were linearly dependent, the equation c₁ ⋅ f₁(x) + c₂ ⋅ f₂(x) + c₃ ⋅ f₃(x) = 0 would represent a horizontal line at y = 0. However, in this case, the functions are linearly independent, so there is no way to combine them to produce the zero function. The graphical representation would show three distinct functions with no possibility of collapsing to y = 0.
Conclusion
The functions f₁(x) = x, f₂(x) = sin(x), and f₃(x) = eˣ are linearly independent because there are no constants c₁, c₂, c₃ (not all zero) that can make the equation c₁ ⋅ f₁(x) + c₂ ⋅ f₂(x) + c₃ ⋅ f₃(x) = 0 true for all x. This means they cannot be combined to produce the zero function y = 0.
Definitions and Concepts
- Linear Dependence (LD): A set of functions is linearly dependent if there exist constants c₁, c₂, c₃ (not all zero) that can make the equation c₁ ⋅ f₁(x) + c₂ ⋅ f₂(x) + c₃ ⋅ f₃(x) = 0 true for all x.
Example of Three Linearly Dependent Functions
- The set of functions is represented as {f₁, f₂, f₃}.
- f₁(x) = x
- f₂(x) = 2x
- f₃(x) = -3x
Explanation
- Write the equation for linear dependence: c₁ ⋅ f₁(x) + c₂ ⋅ f₂(x) + c₃ ⋅ f₃(x) = 0.
- Choose constants c₁ = 1, c₂ = -1, c₃ = 1.
- Substitute the functions and constants into the equation: 1 ⋅ x – 1 ⋅ 2x + 1 ⋅ (-3x) = 0 → x – 2x – 3x = 0 → -4x = 0.
- The equation -4x = 0 is true for all x, so the functions are linearly dependent.
Graphical Interpretation
The fact that these functions can be combined linearly to produce the zero function (a horizontal line at y = 0) means they are dependent on each other. The graph includes the line y = 0 to represent the zero function that results from the linear combination of the functions.
Conclusion
The functions {f₁(x) = x, f₂(x) = 2x, f₃(x) = -3x} are linearly dependent because there exist constants c₁ = 1, c₂ = -1, c₃ = 1 that can make the equation c₁ ⋅ f₁(x) + c₂ ⋅ f₂(x) + c₃ ⋅ f₃(x) = 0 true for all x. This means they can be combined to produce the zero function y = 0, as shown in the graph.
Observation: Constant Multiples of x
In the example of linearly dependent functions {f₁(x) = x, f₂(x) = 2x, f₃(x) = -3x}, we can observe that each function is a constant multiple of the variable x. This relationship plays a key role in their linear dependence.
Explanation
- f₁(x) = x: This function is the base function, representing x itself.
- f₂(x) = 2x: This function is 2 times the base function f₁(x).
- f₃(x) = -3x: This function is -3 times the base function f₁(x).
Since f₂(x) and f₃(x) are constant multiples of f₁(x), they are linearly dependent on f₁(x). This means that there exist constants that can combine these functions to produce the zero function y = 0.
Graphical Interpretation
The graph of these functions would show parallel lines, each representing a different constant multiple of x. The fact that they are constant multiples of each other means that they can be combined in a way that cancels each other out, resulting in the zero function y = 0.
Conclusion
The observation that the functions {f₁(x) = x, f₂(x) = 2x, f₃(x) = -3x} are constant multiples of x explains their linear dependence. This relationship allows them to be combined in a specific way to produce the zero function, a key characteristic of linearly dependent functions.
1. Uniqueness of the General Solution
Example: Consider the differential equation y” – y = 0. The general solution is given by:
y(x) = c₁ * eˣ + c₂ * e^(-x)
1. Uniqueness of the General Solution
Example: If the solutions were linearly dependent, such as f(x) = eˣ and g(x) = 2eˣ, the general solution would be:
y(x) = c₁ * eˣ + c₂ * 2eˣ = (c₁ + 2c₂) * eˣ
Implication: This general solution is redundant because it effectively has only one independent solution. It fails to capture the full complexity of the system, leading to a loss of uniqueness in the general solution.
Why Does This Matter? An Example:
Consider a scenario where the initial conditions are y(0) = 1 and y'(0) = 1. With the linearly dependent solutions f(x) = eˣ and g(x) = 2eˣ, the general solution becomes y(x) = (c₁ + 2c₂) * eˣ.
Now, if we try to find the constants c₁ and c₂ that satisfy the initial conditions, we encounter a problem. Plugging in x = 0, we get the equation c₁ + 2c₂ = 1. However, this single equation has infinitely many solutions for c₁ and c₂.
This lack of uniqueness means that there are numerous combinations of c₁ and c₂ that satisfy the equation c₁ + 2c₂ = 1. As a result, we can’t uniquely determine the constants, leading to a multitude of possible solutions for y(x) that satisfy the given initial conditions.
Numerical Examples:
Let’s explore corrected numerical examples to highlight the lack of uniqueness:
- Example 1: If c₁ = 1 and c₂ = -0.5, the equation c₁ + 2c₂ = 1 is satisfied. This leads to y(x) = eˣ – eˣ = 0.
- Example 2: If c₁ = 0 and c₂ = 0.5, the equation c₁ + 2c₂ = 1 is also satisfied. This leads to y(x) = 0 + 0.5 * 2eˣ = eˣ.
Both cases result in different functions, demonstrating the non-uniqueness of the solutions.