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Chapter 8: Problem 47

Prove the Cauchy-Schwarz Inequality: $$|\mathbf{a} \cdot \mathbf{b}| \leq|\mathbf{a}||\mathbf{b}|$$

Short Answer

Expert verified
The Cauchy-Schwarz inequality is proved using the non-negativity of squared expressions.

Step by step solution

01

Understanding the Vectors

Consider two vectors \( \mathbf{a} \) and \( \mathbf{b} \) in a vector space with the dot product \( \mathbf{a} \cdot \mathbf{b} \), which can also be expressed as \( |\mathbf{a}||\mathbf{b}| \cos \theta \), where \( \theta \) is the angle between them.
02

Expression Simplification with Dot Product

The Cauchy-Schwarz inequality states that \( |\mathbf{a} \cdot \mathbf{b}| \leq |\mathbf{a}||\mathbf{b}| \). Expanding the dot product in terms of its components, we have \( (a_1b_1 + a_2b_2 + \ldots + a_nb_n)^2 \leq (a_1^2 + a_2^2 + \ldots + a_n^2)(b_1^2 + b_2^2 + \ldots + b_n^2) \). This involves the expression of the dot product of \( \mathbf{a} \) and \( \mathbf{b} \) as a sum of products of their respective components.
03

Constructing a Square Expression

Examine the expression \( |\mathbf{a} \cdot \mathbf{b}|^2 \). Consider the squared magnitude of the dot product: \( (a_1b_1 + a_2b_2 + \ldots + a_nb_n)^2 \). This can be expanded as \( a_1^2b_1^2 + a_1b_1a_2b_2 + \ldots + a_nb_n^2 \), which involves multiple terms of products.
04

Introducing a Different Perspective

Introduce the vector \( \mathbf{c} = \mathbf{a} - \lambda \mathbf{b} \) where \( \lambda \) is a scalar. The vector was designed such that its dot product with itself is non-negative: \( \mathbf{c} \cdot \mathbf{c} \geq 0 \).
05

Expanding the Expression

Now expand the expression \( (\mathbf{a} - \lambda \mathbf{b}) \cdot (\mathbf{a} - \lambda \mathbf{b}) \). This yields: \( \mathbf{a} \cdot \mathbf{a} - 2\lambda (\mathbf{a} \cdot \mathbf{b}) + \lambda^2 (\mathbf{b} \cdot \mathbf{b}) \geq 0 \). This quadratic in \( \lambda \) should have no real roots.
06

Comparing Quadratic Relation

Evaluate the discriminant of the quadratic: \( (-2 \mathbf{a} \cdot \mathbf{b})^2 - 4(\mathbf{a} \cdot \mathbf{a})(\mathbf{b} \cdot \mathbf{b}) \leq 0 \), which simplifies to \( (\mathbf{a} \cdot \mathbf{b})^2 \leq (\mathbf{a} \cdot \mathbf{a})(\mathbf{b} \cdot \mathbf{b}) \). The non-positive discriminant confirms the lack of real roots.
07

Taking Square Roots

By taking the square root of both sides, we conclude \( |\mathbf{a} \cdot \mathbf{b}| \leq |\mathbf{a}||\mathbf{b}| \). This confirms the geometry-based interpretation of the Cauchy-Schwarz inequality with respect to angle projections.

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Key Concepts

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

Vector Space
In mathematics, a vector space is a collection of objects called vectors that can be added together and multiplied by numbers, which are called scalars in this context. Vector spaces are fundamental in linear algebra and provide a framework for operations involving vectors.
Key properties of vector spaces include:
  • Vectors can be added together to yield another vector.
  • Vectors can be multiplied by scalars, transforming the vector.
  • Every vector space contains a zero vector, which acts as an additive identity.
  • Adding a vector to its negative results in the zero vector.
In the context of the Cauchy-Schwarz Inequality, consider two vectors \( \mathbf{a} \) and \( \mathbf{b} \). These vectors exist in a vector space where we can use operations like addition and dot product to explore relationships between them.
Dot Product
The dot product, also known as the scalar product, is a way to multiply two vectors that results in a scalar (a single number). It is a crucial operation in vector algebra, often used to measure angles and lengths within a vector space.
The dot product has the following characteristics:
  • It is calculated by multiplying corresponding components of the two vectors and then summing the results.
  • For vectors \( \mathbf{a} = (a_1, a_2, \, \ldots, a_n) \) and \( \mathbf{b} = (b_1, b_2, \, \ldots, b_n) \), the dot product is \( a_1b_1 + a_2b_2 + \, \ldots + a_nb_n \).
  • The dot product is related to the cosine of the angle \( \theta \) between the vectors: \( \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}| \cos \theta \).
  • It is commutative, meaning \( \mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a} \).
Understanding the dot product is essential in proving the Cauchy-Schwarz Inequality as it lays the foundation for interpreting the relationship between vectors.
Quadratic Expression
A quadratic expression is a polynomial of the form \( ax^2 + bx + c \) where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable. Quadratic expressions are central to various mathematical problems and often appear in the context of quadratic equations.
Here are some important points about quadratic expressions:
  • They can be factored or expanded using techniques like completing the square or using the quadratic formula.
  • The graph of a quadratic expression is a parabola, opening upwards if \( a > 0 \) and downwards if \( a < 0 \).
  • The expression \( x^2 - 2\lambda (\mathbf{a} \cdot \mathbf{b}) + \lambda^2 (\mathbf{b} \cdot \mathbf{b}) \) in the Cauchy-Schwarz proof is a quadratic in \( \lambda \).
  • Studying the discriminant, \( b^2 - 4ac \), can reveal the nature of the roots of a quadratic equation.
In proving the Cauchy-Schwarz Inequality, constructing a suitable quadratic expression is key to showing that its solution space does not allow real roots, implying an important geometric interpretation.
Inequalities
Inequalities are mathematical statements indicating that one value is less than or greater than another. They play a crucial role in mathematical analysis and are often used to establish unknown relationships between variables.
Some fundamental properties of inequalities include:
  • Inequalities express a range of solutions rather than a fixed number.
  • They are used to prove bounds such as \( x \geq 0 \) or \( y < 10 \).
  • The Cauchy-Schwarz Inequality specifically states \( |\mathbf{a} \cdot \mathbf{b}| \leq |\mathbf{a}||\mathbf{b}| \).
  • Understanding inequalities involves knowing when they are strict (e.g., \( < \)) or non-strict (e.g., \( \leq \)).
  • Manipulating inequalities requires care, especially when multiplying or dividing by negative numbers.
The Cauchy-Schwarz Inequality elegantly illustrates how inequalities can convey geometric relationships in vector spaces, showing how projections within these spaces are bounded.

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Most popular questions from this chapter

Determine whether or not \(\mathbf{x}\) is an eigenvector of \(A .\) If it is, determine its associated eigenvalue. $$(\mathrm{a})A=\left[ \begin{array}{rr}{3} & {-1} \\ {2} & {0}\end{array}\right] \quad \mathbf{x}=\left[ \begin{array}{l}{1} \\\ {2}\end{array}\right]$$ $$(\mathrm{b})A=\left[ \begin{array}{rrr}{-3} & {-1} & {5} \\ {-2} & {1} & {2} \\ {-2} & {-1} & {4}\end{array}\right] \quad \mathbf{x}=\left[ \begin{array}{l}{2} \\ {1} \\ {1}\end{array}\right]$$ $$(\mathrm{c})A=\left[ \begin{array}{rr}{1} & {-1} \\ {-2} & {0}\end{array}\right] \quad \mathbf{x}=\left[ \begin{array}{l}{1} \\\ {0}\end{array}\right]$$ $$(\mathrm{d}) A=\left[ \begin{array}{rr}{1} & {2} \\ {-2} & {1}\end{array}\right] \quad \mathbf{x}=\left[ \begin{array}{l}{2} \\ {2 i}\end{array}\right]$$ $$(\mathrm{e}) A=\left[ \begin{array}{ll}{2+3 a} & {-2-2 a} \\ {3+3 a} & {-3-2 a}\end{array}\right] \quad \mathbf{x}=\left[ \begin{array}{l}{2} \\\ {3}\end{array}\right]$$ $$(\mathrm{f}) A=\left[ \begin{array}{rrr}{-9} & {4} & {6} \\ {-6} & {3} & {4} \\ {-9} & {4} & {6}\end{array}\right] \quad \mathbf{x}=\left[ \begin{array}{l}{2} \\ {0} \\ {3}\end{array}\right]$$

Darwin's finches have been used to study how differences in bird morphology are related to differences in diet. Morphological measurements (in mm) of three species are given in the table for three traits. $$\begin{array}{|c|c|c|c|c|}\hline \text { Species } & {\text { Wing length }} & {\text { Tarsus length }} & {\text { Beak length }} \\ \hline G . \text { difficilis } & {64} & {18.1} & {9.6} \\ {G . \text { fulliginosa }} & {62.1} & {17.9} & {9.6} \\ {\text {G. scandens}} & {73.1} & {21.1} & {14.5} \\\ \hline\end{array}$$ The proportion of time spent feeding on different types of food for these three species is given in the following table. $$\begin{array}{|c|c|c|c|}\hline \text { Species } & {\text { Seeds }} & {\text { Pollen }} & {\text { Other }} \\ \hline G . \text { difficilis } & {0.67} & {0.23} & {0.1} \\ {G . \text { fuliginosa }} & {0.7} & {0.28} & {0.02} \\ \hline G . \text { scandens } & {0.14} & {0} & {0.86} \\\ \hline\end{array}$$ (a) Thinking of the morphology of each species as a point in \(\mathbb{R}^{3},\) calculate the morphological distance between each pair of species. (b) Thinking of the diet of each species as a point in \(\mathbb{R}^{3}\) , calculate the diet distance between each pair of species. (c) Do species that are morphologically most similar also tend to have the most similar diets?

Find the unit vector that points in the same direction as the given vector and express it in terms of the standard basis vectors. $$\begin{array}{ll}{\text { (a) } \mathbf{i}+\mathbf{j}} & {\text { (b) } \mathbf{i}+\mathbf{j}+\mathbf{k}} \\ {\text { (c) } 2 \mathbf{i}-\mathbf{k}} & {\text { (d) } 4 \mathbf{i}+6 \mathbf{j}-\mathbf{k}}\end{array}$$

Express the solution to the recursion \(\mathbf{n}_{t+1}=A \mathbf{n},\) in terms of the eigenvectors and eigenvalues of \(A,\) assuming arbitrary initial conditions. \(A=\left[ \begin{array}{ll}{0} & {1} \\ {1} & {0}\end{array}\right]\)

Construct the matrix diagram for each of the following models. Ecological succession is a process by which a biological community progresses between different states. Suppose the community can be either primarily grassland, primarily shrub, or primarily forest, and this can change from year to year. Using \(g_{t}, s_{t},\) and \(f_{t}\) to denote the probability that the community is in each of the three possible states at time \(t\) we have $$\begin{aligned} g_{t+1} &=0.9 g_{t}+0.001 s_{t} \\ s_{t+1} &=0.1 g_{t}+0.799 s_{t}+0.001 f_{t} \\ f_{t+1} &=0.2 s_{t}+0.999 f_{t} \end{aligned}$$
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