Matrices and Vectors Quiz
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Questions and Answers

What is the primary focus of the first chapter?

  • Sums and integrals
  • Differential equations
  • Matrices and vectors (correct)
  • Complex numbers

The Nash bargaining problem is an example of constrained optimisation.

True (A)

What is the function of an implicit function theorem?

To show conditions under which a relation defines a function.

The _____ theorem of calculus connects differentiation and integration.

<p>Fundamental</p> Signup and view all the answers

Match the following concepts with their correct applications:

<p>Gradient = Direction of steepest ascent Lagrange's method = Optimisation with constraints Partial derivatives = Rate of change with respect to one variable Complex numbers = Use in electrical engineering</p> Signup and view all the answers

Which of the following is NOT a type of function discussed in the content?

<p>Separable functions (D)</p> Signup and view all the answers

All integrals are finite.

<p>False (B)</p> Signup and view all the answers

What is required for the prices ATq to be more attractive than market prices?

<p>The prices ATq must be lower than market prices. (D)</p> Signup and view all the answers

In a zero sum game, the sum of the payoffs to both players is always positive.

<p>False (B)</p> Signup and view all the answers

What is the purpose of shadow prices in linear programming?

<p>To value the stock given the relationship between the amount of stock and the finished goods.</p> Signup and view all the answers

A two person ___ game is one where whatever one player wins, the other loses.

<p>zero sum</p> Signup and view all the answers

Match the concepts to their definitions:

<p>Shadow Prices = Prices that reflect the value of stock in the dual problem Zero Sum Game = A situation where one player's gain is the other's loss Mixed Strategies = Assigning probabilities to various strategies Dual Problem = The relationship between minimum cost and maximum revenue</p> Signup and view all the answers

What does the input vector y represent in the production process?

<p>The required quantity of ingredients (B)</p> Signup and view all the answers

The price vector p lists the sales prices of the commodities only.

<p>False (B)</p> Signup and view all the answers

What is the primary goal of the baker in the linear programming problem?

<p>To maximize revenue from sales.</p> Signup and view all the answers

The constraints in the baker's linear programming problem require that x must not be less than ____.

<p>0</p> Signup and view all the answers

Match the following concepts with their definitions:

<p>Input vector (y) = Required quantities of ingredients Price vector (p) = Prices of commodities Revenue = Income from selling products Linear programming = Optimization method for resource allocation</p> Signup and view all the answers

In a linear programming problem, what does the notation Ax refer to?

<p>The amount of ingredients required for baking x (B)</p> Signup and view all the answers

A baker can freely choose the quantity x of baked goods without any constraints.

<p>False (B)</p> Signup and view all the answers

What is the dual problem relating to the baker's production?

<p>Determining the price vector q for the ingredients being sold.</p> Signup and view all the answers

The equation for revenue generated from selling x is expressed as ____.

<p>pT x</p> Signup and view all the answers

What represents the maximum revenue achievable by the baker?

<p>The outcome of the linear programming problem (C)</p> Signup and view all the answers

What is the result of scalar multiplying a matrix A by a scalar c?

<p>All entries of A are multiplied by c (B)</p> Signup and view all the answers

A matrix with 3 rows and 2 columns is called a 2 × 3 matrix.

<p>False (B)</p> Signup and view all the answers

What is the notation used to express the entry in the second row and first column of a matrix C?

<p>c21</p> Signup and view all the answers

A _____ matrix is obtained by adding corresponding entries of two matrices.

<p>sum</p> Signup and view all the answers

Match the following operations with their descriptions:

<p>Scalar multiplication = Each entry of the matrix is multiplied by a scalar Matrix addition = Adding corresponding entries of two matrices Matrix subtraction = Subtracting corresponding entries of two matrices Zero matrix = A matrix with all entries equal to zero</p> Signup and view all the answers

If matrix C is a 2 × 3 matrix and matrix D is also a 2 × 3 matrix, what will be the dimensions of the resulting matrix when C is added to D?

<p>2 × 3 (A)</p> Signup and view all the answers

The scalars used in matrix operations can only be real numbers.

<p>False (B)</p> Signup and view all the answers

How does one denote the zero matrix?

<p>0</p> Signup and view all the answers

A matrix with m rows and n columns is referred to as an _____ matrix.

<p>m × n</p> Signup and view all the answers

What does the vector product of two vectors u and v represent in a geometric context?

<p>A vector orthogonal to both u and v (D)</p> Signup and view all the answers

The direction of the vector product v × u is the same as u × v.

<p>False (B)</p> Signup and view all the answers

What is the formula for the area of the parallelogram formed by vectors u and v?

<p>u × v</p> Signup and view all the answers

The vector product, also known as the ______, is defined as the vector orthogonal to both u and v.

<p>cross product</p> Signup and view all the answers

Match the following terms with their definitions:

<p>u × v = Area of the parallelogram formed by u and v u · v = Dot product of u and v det P = Determinant of matrix P u × v length = Magnitude of the vector product u and v</p> Signup and view all the answers

Which formula correctly gives the area of the parallelepiped formed by vectors u, v, and w?

<p>Both B and C (B)</p> Signup and view all the answers

The expression for the vector product can be simplified using determinants.

<p>True (A)</p> Signup and view all the answers

What is the significance of the right hand rule in vector products?

<p>It determines the direction of the vector product.</p> Signup and view all the answers

The length of the vector product u × v is equal to the modulus of the ______.

<p>determinant</p> Signup and view all the answers

Which of the following statements about the equations of a plane is true?

<p>A plane can be defined using two non-parallel vectors. (B)</p> Signup and view all the answers

Flashcards

ISBN

International Standard Book Number. A unique 13-digit number used to identify books worldwide.

Linear Programming

A mathematical technique used for optimizing a linear objective function subject to linear constraints.

Quadric Surfaces

3D surfaces described by quadratic equations. Examples include spheres, paraboloids, and hyperboloids.

Tangent Plane

A plane that touches a 3D surface at a single point and has the same slope as the surface at that point.

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Gradient

A vector that indicates the direction of greatest increase of a function at a point.

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Positive/Negative Definite (Quadratic Forms)

A quadratic form is positive definite if it is always positive for any non-zero input vector. It is negative definite if it is always negative for any non-zero input vector.

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Constrained Optimisation

Finding the optimal value of a function subject to one or more constraints or limitations.

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Matrix

A rectangular array of numbers organized in rows and columns. Matrices are used to represent and manipulate information efficiently.

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m × n matrix

A matrix with 'm' rows and 'n' columns. The first number represents the number of rows, and the second number represents the number of columns.

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Scalar

A single number, often used as a building block of matrices. It can be multiplied with a matrix to scale its values.

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Scalar Multiplication

The process of multiplying a matrix by a scalar. Every entry of the matrix is multiplied by the scalar.

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Matrix Addition

Adding two matrices of the same dimensions by adding corresponding entries.

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Matrix Subtraction

Subtracting two matrices of the same dimensions by subtracting corresponding entries.

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Zero Matrix

A matrix where all entries are zero. It acts as the additive identity in matrix addition.

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Additive Identity

A zero matrix added to any other matrix results in the same matrix. It's like adding zero to a number.

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Matrix Equation

A mathematical equation where matrices are involved. Used to solve systems of linear equations and represent relationships between variables.

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Dual Problem

A linear programming problem that seeks to minimize the cost of acquiring a baker's ingredients, considering the resources needed to produce finished goods and sell them at given market prices.

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Shadow Prices

The optimal prices for ingredients, determined by solving the dual problem, which reflect their value in producing finished goods and selling them at market prices.

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Zero-Sum Game

A game where the total gains of one player exactly equal the total losses of the other player.

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Payoff Matrix

A table representing the possible outcomes of a game, where rows represent strategies for one player and columns represent strategies for the other player.

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Mixed Strategies

Players' actions in a game where they assign probabilities to each of their strategies, rather than choosing a single one.

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Vector product

A binary operation on two vectors in three-dimensional space, resulting in a vector orthogonal to both input vectors.

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Plane equation

An equation that defines all points lying on a specific plane in three-dimensional space.

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Right-hand rule

A convention used to determine the direction of the vector product of two vectors. If you curl the fingers of your right hand from the first vector to the second, your thumb points in the direction of the cross product.

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Normal vector

A vector perpendicular to a plane, indicating the plane's orientation.

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Parallelogram area

The area of a parallelogram formed by two vectors equals the magnitude of their cross product.

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Determinant (2x2)

The determinant of a 2x2 matrix represents the area of the parallelogram formed by its column vectors.

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Parallelepiped volume

The volume of a parallelepiped is equal to the absolute value of the determinant of the 3x3 matrix formed by its adjacent sides.

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Cartesian equation of a plane

An equation of a plane expressed in terms of the x, y, and z coordinates. It can be derived from the parametric form using the vector product.

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Vector equation of a plane

An equation of a plane expressed in terms of a point on the plane and a normal vector. It's equivalent to the Cartesian form.

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Geometric interpretation of a determinant

The determinant of a square matrix represents the volume (or area in 2D) of the parallelepiped (or parallelogram in 2D) formed by its column vectors.

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What is a price vector?

A price vector lists the prices at which different commodities can be bought or sold. It's a way of representing the cost of each item in a bundle.

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How is the value of a commodity bundle calculated?

The value of a commodity bundle is calculated by multiplying the price vector (p) by the commodity vector (x). This gives you the total cost or revenue of the bundle.

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What is a hyperplane?

A hyperplane is a mathematical concept representing a flat surface in n-dimensional space. It separates the space into two regions. In the context of price vectors, hyperplanes represent points where the total cost of a bundle is equal to a specific value.

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How is the revenue from selling a baked good calculated?

The revenue from selling a baked good is calculated by multiplying the price vector (p) by the quantity of baked goods (x). This gives the total amount of money earned from selling the baked goods.

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What is the constraint in the baker's problem?

The constraint is the restriction on the amount of baked goods the baker can produce. This is limited by the amount of available ingredients. The baker must ensure the required ingredients (y = Ax) do not exceed the available stock of ingredients (b).

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What is a linear programming problem?

A linear programming problem involves finding the best solution (usually maximizing a function) within a set of constraints that are expressed as linear equations or inequalities.

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What is the dual problem in the context of the baker's problem?

The dual problem is the perspective of a factory that wants to buy the baker's ingredients. The factory wants to determine the best price vector (n) to offer the baker to secure the ingredients.

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How does the dual problem relate to the baker's revenue?

The dual problem helps determine the price the factory should offer the baker for the ingredients. The factory wants to buy the ingredients at a price that ensures the baker is better off selling the ingredients than using them to bake goods and selling those.

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Study Notes

Matrices and Vectors

  • A matrix is a rectangular array of numbers, used for systematic calculations.
  • An m × n matrix has m rows and n columns.
  • Entries are called scalars, usually real numbers.
  • Scalar multiplication multiplies each entry of a matrix by a scalar.
  • Matrix addition adds corresponding entries.
  • Matrix subtraction subtracts corresponding entries.
  • The zero matrix has all entries equal to zero.
  • Vectors can be represented in 2D, 3D and n dimensions.
  • Lines and planes can be represented using vectors.
  • The vector product (or cross product) is orthogonal to both vectors.
  • The area of a parallelogram is given by the magnitude of the vector product.
  • Determinants of 2x2 and 3x3 matrices can represent areas and volumes respectively.
  • Commodity bundles, linear production models, price vectors, and linear programming examples are mentioned.

Functions of One Variable

  • Functions of one variable can be elementary functions (power, exponential, trigonometric).
  • Functions can be combined (sum, difference, product, quotient).
  • Inverse functions reverse the action of a function.
  • Derivatives are measures of instantaneous rate of change.
  • Higher-order derivatives are successive derivatives.
  • Taylor series approximates a function using derivatives.
  • Conic sections (parabola, ellipse, hyperbola, circle) are described through implicit equations.

Functions of Several Variables

  • Functions of two variables are defined, including linear and affine functions.
  • Quadric surfaces are mentioned.
  • Partial derivatives measure rate of change with respect to one variable (others held constant).
  • Tangent planes approximate functions near a point.
  • The gradient is a vector representing all the partial derivatives.
  • Directional derivatives measure the rate of change along a specific direction.
  • Functions of more than two variables are also discussed, including Tangent hyperplanes and Directional derivatives.
  • Indifference curves, profit maximization, and contract curve applications are mentioned

Stationary Points

  • Stationary points for single-variable functions are critical points.
  • Optimization problems find the maximum or minimum of a function.
  • Constrained optimisation involves finding the extrema of a function subject to constraints.
  • The use of computer systems is suggested for calculations.
  • Stationary points, gradients, and stationary points for functions of more than two variables are mentioned in general.

Vector Functions

  • Vector-valued functions output vectors.
  • Affine functions and flats relate to lines, planes, and higher-dimensional spaces.
  • Derivatives of vector-valued functions are described.
  • Chain rule applies to derivatives of composite functions.
  • Second derivatives and a Taylor series for scalar valued functions of n variables are included in the study material.

Optimisation of Scalar Valued Functions

  • Change of basis and quadratic forms are discussed.
  • Definite matrices are mentioned.
  • Applications include the Nash bargaining problem, inventory control, least squares analysis, Kuhn–Tucker conditions, linear programming, and saddle points.

Inverse Functions

  • Local inverses of scalar-valued functions are shown in detail, and their differentiability.
  • Inverse trigonometric functions are discussed.
  • Local inverses of vector-valued functions, coordinate systems (e.g., polar coordinates), differential operators are also included.

Implicit Functions

  • Implicit differentiation is explained.
  • Implicit functions, Implicit function theorem, and shadow prices are presented.

Differentials

  • Matrix algebra and linear systems are mentioned.
  • Differentials are used to approximate small changes in a function.
  • Applications include Slutsky equations.

Sums and Integrals

  • Sums, integrals, fundamental theorem of calculus, notations, and standard integrals are explained.
  • Techniques like partial fractions, completing the square, change of variables, and integration by parts are included.
  • Infinite sums and integrals, dominated convergence, differentiating integrals, power series are included in this chapter.
  • This chapter also covers applications like Probability, probability density functions, binomial distribution, Poisson distribution, Normal distribution, and sums of random variables.

Multiple Integrals

  • Repeated integrals evaluate multidimensional integrals over intervals.
  • Applications such as Joint probability distributions, Marginal probability distributions, Expectation, Variance, Covariance, Independent random variables, Generating functions, and Multivariate normal distributions are mentioned.

Differential Equations of Order One

  • Differential equations and their solutions are explained.
  • General solutions and boundary conditions of ordinary equations.
  • Separable, exact, linear, homogeneous equations, and related solutions are included.
  • Partial differential equations, exact equations, and change of variable in partial differential equations are included in this section.

Complex Numbers

  • Complex numbers and their use in solving quadratic equations are discussed.
  • Modulus, argument, complex roots, and polynomials are included in this section.
  • Applications like Characteristic functions and Central limit theorem are discussed.

Linear Differential and Difference Equations

  • Linear differential equations and difference equations, their methods of solutions, linear operators, and convergence, along with corresponding applications.
  • Applications include Cobweb models, Gambler’s ruin, Systems of linear equations.

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