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Chapter 6: Problem 6

Test the sets of polynomials for linear independence. For those that are linearly dependent, express one of the polynomials as a linear combination of the others. $$\left\\{1+x, 1+x^{2}, 1-x+x^{2}\right\\} \text { in } \mathscr{P}_{2}$$

Short Answer

Expert verified
The set of polynomials is linearly independent.

Step by step solution

01

Understand the Problem

We need to determine if the given set of polynomials \( \{1+x, 1+x^{2}, 1-x+x^{2}\} \) in \( \mathscr{P}_{2} \) are linearly independent. If they are dependent, we must express one polynomial as a linear combination of the others.
02

Set Up the Linear Combination

Assume that a linear combination of these polynomials equals zero: \( a(1+x) + b(1+x^2) + c(1-x+x^2) = 0 \) for some scalars \( a, b, c \).
03

Expand and Combine Like Terms

Expand and combine like terms to get: \( (a+b+c) + ax - cx + (b+c)x^2 = 0 \). This must hold for all values of \( x \), so each coefficient must equal zero.
04

Set Up Equations for Coefficients

From the linear combination, we have the system of equations: \[ a + b + c = 0 \] \[ a - c = 0 \] \[ b + c = 0 \].
05

Solve the System of Equations

Using the equation \( a - c = 0 \), we have \( a = c \). Substitute this into \( b + c = 0 \) to get \( b = -c \). Substitute into \( a + b + c = 0 \) to find \( c = 0 \), and thus \( a = 0 \), \( b = 0 \).
06

Interpret the Results

Since the only solution is the trivial solution \( a = 0 \), \( b = 0 \), \( c = 0 \), the polynomials are linearly independent. Therefore, no polynomial can be expressed as a linear combination of the others.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Combination
A linear combination is a mathematical concept where we express a function or vector as a sum of multiples of other functions or vectors. In our exercise, we examine if a polynomial can be written as a linear combination of other polynomials. Specifically, we check if coefficients exist such that:
  • \( a(1+x) + b(1+x^2) + c(1-x+x^2) = 0 \)
Where \( a, b, c \) are scalars. Each polynomial is multiplied by a scalar, and together, they should sum up to zero.
This makes a linear combination an essential tool for determining the linear independence of a set of functions or vectors.
Key steps in understanding linear combinations:
  • Express the set functions as a weighted sum.
  • Solve for the coefficients (scalars) that make this sum equal zero.
  • Identify if only a trivial solution (all scalars equal zero) exists.
Polynomials
Polynomials are expressions made up of variables and coefficients, where the variables are raised to whole-number exponents. In our exercise, we deal with polynomials like \( 1+x \), \( 1+x^2 \), and \( 1-x+x^2 \) within the polynomial space \( \mathscr{P}_{2} \).
Understanding polynomials involves knowing:
  • Terms: Individual parts of polynomials, such as \( x^2 \).
  • Coefficients: Numbers multiplying the variables, in expressions like \( 1 \) in \( 1+x \).
  • Degree: The highest exponent of the variable, indicating the polynomial's order.
In this exercise's context, you need to manipulate these terms to analyze their relationships, like finding their part in a sequence or determining equations that equal zero when combined.
Given polynomials help us explore wider mathematical concepts like independence and dependence in a polynomial context.
System of Equations
A system of equations consists of multiple equations that are solved together, which means finding values for variables that satisfy all the equations simultaneously. During the exercise, we arrange a system from the coefficients of the polynomials to explore their independence:
  • \( a + b + c = 0 \)
  • \( a - c = 0 \)
  • \( b + c = 0 \)
These equations arise from the linear combination process and are critical in deciding linear independence.
To solve a system of equations, follow these steps:
  • Isolate variables using substitution or elimination methods.
  • Simplify until you deduce the values of variables that meet all equations.
  • Identify the solution types 鈥� notably, the trivial solution suggests independence.
In essence, if the only solution to this system is that all variables equal zero, it claims the set is linearly independent and no single polynomial can be represented by the others.

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Most popular questions from this chapter

Find either the nullity or the rank of \(T\) and then use the Rank Theorem to find the other. \(T: M_{22} \rightarrow M_{22}\) defined by \(T(A)=A B,\) where \(B=\left[\begin{array}{rr}1 & -1 \\ -1 & 1\end{array}\right]\)

Find the solution of the differential equation that satisfies the given boundary condition(s). $$x^{\prime}+x=0, x(1)=1$$

A strain of bacteria has a growth rate that is proportional to the size of the population. Initially, there are 100 bacteria; after 3 hours, there are 1600 (a) If \(p(t)\) denotes the number of bacteria after hours, find a formula for \(p(t)\) (b) How long does it take for the population to double? (c) When will the population reach one million?

Determine whether the linear transformation T is (a) one-to-one and (b) onto. $$T: \mathbb{R}^{3} \rightarrow \mathbb{M}_{22} \text { defined by } T\left[\begin{array}{l}a \\\b \\\c\end{array}\right]=\left[\begin{array}{ll} a-b & b-c \\\a+b & b+c\end{array}\right]$$

Find a basis for \(\operatorname{span}\left(1-2 x, 2 x-x^{2}, 1-x^{2}, 1+x^{2}\right)\) in \(\mathscr{P}_{2}\).
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