Linear Algebra Chapter 2: Systems of Linear Equations
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Questions and Answers

What is a system of linear equations?

  • Any set of equations that includes at least one linear equation.
  • A collection of linear equations that can be solved using substitution or elimination methods. (correct)
  • A group of equations that all share the same solution.
  • A set of equations where each equation is linear and dependent on a specific variable.

Which of the following describes an augmented matrix?

  • A matrix that includes both the coefficients and the constants from the equations. (correct)
  • A matrix derived only from nondegenerate linear equations.
  • A matrix that contains only the coefficients of a linear system.
  • A matrix representation of the solutions to a linear system of equations.

What is a leading variable in a nondegenerate linear equation?

  • A variable that does not appear in the solution set.
  • The variable that can be isolated to express other variables in terms of it. (correct)
  • A variable that can take any value in its domain.
  • A variable that appears as a coefficient in all transformations.

Which term describes linear equations that do not have a unique solution?

<p>Dependent equations. (D)</p> Signup and view all the answers

Which elementary operation on equations does not alter the solution set?

<p>Adding a multiple of one equation to another equation. (D)</p> Signup and view all the answers

What is the first step when using back-substitution to find the solution of a system of equations?

<p>To determine the free variables and assign them arbitrary values (D)</p> Signup and view all the answers

What is the value of the pivot variable $x_3$ when $x_4 = b$ in the back-substitution example?

<p>$7(b - 1)$ (A)</p> Signup and view all the answers

Which of the following correctly represents $x_1$ in terms of the free variables?

<p>$10b - a - 9$ (A)</p> Signup and view all the answers

In which form is the Gaussian elimination algorithm described as being simpler and faster?

<p>Matrix form (C)</p> Signup and view all the answers

What operation is performed on $L_2$ to eliminate $x_1$ during forward elimination?

<p>$L_2 = L_2 - 2L_1$ (D)</p> Signup and view all the answers

In the system of equations presented, how many variables are present?

<p>3 (C)</p> Signup and view all the answers

What is the purpose of the free variables in the back-substitution process?

<p>To allow for an infinite number of solutions (A)</p> Signup and view all the answers

Which equation closely resembles the structure of $x_1$ after back-substitution is performed?

<p>$x_1 + x_2 - 2x_3 + 4x_4 = 5$ (D)</p> Signup and view all the answers

What does it indicate when a system of equations results in a degenerate equation with a non-zero constant?

<p>The system has no solution. (B)</p> Signup and view all the answers

In the Gaussian elimination process, which variable is chosen as the pivot for the first equation?

<p>x1 (D)</p> Signup and view all the answers

After the first step of forward elimination, what form does the second equation take?

<p>x3 - 7x4 = -7 (D)</p> Signup and view all the answers

What is the result of removing x1 from L3 when applying Gaussian elimination?

<p>0x3 + 0x4 = 0 (B)</p> Signup and view all the answers

During the Gaussian elimination process, how is x1 removed from L2?

<p>By setting m = -a1,1 = -2 and replacing L2 with L2 - 2L1. (D)</p> Signup and view all the answers

What condition must be met for a system of equations to have infinitely many solutions?

<p>At least one equation is exactly the linear combination of others. (A)</p> Signup and view all the answers

What happens to the system of equations if a pivot variable cannot cancel out other variables successfully?

<p>The system may become inconsistent and thus have no solution. (D)</p> Signup and view all the answers

After completing the forward elimination, what does the presence of a row with all zeros suggest?

<p>The system has infinitely many solutions. (C)</p> Signup and view all the answers

What is the unique solution for x3 in the provided system of equations?

<p>2 (B)</p> Signup and view all the answers

What is the value of x2 based on the derived equations?

<p>-3 (D)</p> Signup and view all the answers

What defines an echelon matrix based on the properties stated?

<p>Leading nonzero entries in rows are to the right of preceding rows. (A)</p> Signup and view all the answers

Which of the following describes the pivots of the echelon matrix?

<p>They must be the first nonzero entry in their respective rows. (C)</p> Signup and view all the answers

According to the definition of row canonical form, what is a requirement?

<p>All leading coefficients must be 1. (A)</p> Signup and view all the answers

In the example given, what columns were identified as containing pivots?

<p>C2, C4, C6, C7 (A)</p> Signup and view all the answers

Which condition is not a property of an echelon matrix?

<p>Leading nonzero entries in a row must be directly below each other. (B)</p> Signup and view all the answers

From the equations provided, what is the final value of x1?

<p>1 (A)</p> Signup and view all the answers

What is the general solution of the system Ax = b?

<p>x* = [3 - t, 2t - 5, t] (C)</p> Signup and view all the answers

Which variable is considered free in this system of equations?

<p>x3 (C)</p> Signup and view all the answers

What operation is performed on R3 in the process of transforming the matrix?

<p>R3 ← R3 - R2 (C)</p> Signup and view all the answers

What is the value of x2 when t = 0 in the particular solution?

<p>-5 (A)</p> Signup and view all the answers

What does the notation x* = xp + xh represent in this context?

<p>The sum of particular and homogeneous solutions (B)</p> Signup and view all the answers

Which row operation was first applied to R2 during matrix reduction?

<p>R2 ← 3R2 - R1 (D)</p> Signup and view all the answers

What is the relationship between the parameters t and the free variable x3?

<p>t is equal to x3 (B)</p> Signup and view all the answers

What does the equation 3x1 + x2 + x3 = 4 simplify to when substituting for x2?

<p>3x1 + 2t - 5 + t = 4 (D)</p> Signup and view all the answers

What is the value of $x^_1$ when $xp = x^_0 = egin{bmatrix} -1 \ 0 \ 0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -1 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0 }}$ at $t=1$?

<p>-1 (C)</p> Signup and view all the answers

What is the value of $t$ when calculating $x^*_2 = xp + egin{bmatrix} -1 \ 0 \ \ \ \ \ \ 2 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -2 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -1\1\end{bmatrix} t$?

<p>1 (B)</p> Signup and view all the answers

What is the outcome of the calculation $x^*_2 = xp + egin{bmatrix} -1 \ 0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2 \ \ \ \ \ \ \ -1 \ \ \ \ \ \ \ \ 2 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \s \ \ \ \ \ \ \ \ 1\ \ \ \ \ \ \ \ \ \ \ \ 2 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2 \ \ \ \ \ \ \ \ \ -1 \\ -1 \ 0\\-1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -4 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -2 \end{bmatrix} t$?

<p>-4 (C)</p> Signup and view all the answers

Which pair represents the correct values of $xp$ and $t$ for the calculation of $x^*_0$?

<p>$xp = egin{bmatrix} -2 \ 0 \ 1 \ _\ \ _\ \ _\ \ _\ \ _\ \ _\ \ _\ \ _\ \ _\ \ _\ \ _\ \ _\ \ \ \ \ \ -1 \ 0} , t=1$ (C)</p> Signup and view all the answers

What is value of $x^*_2$ when $xp = egin{bmatrix} -1 \ 0 \ -1 \ 1 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -2 \ 0 \ \ \ \ -1 \ 2} $ and $t=1$?

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

What is the main purpose of using both $xp$ and calculations involving $t$ in the given equations?

<p>To determine a new state based on previous values. (C)</p> Signup and view all the answers

When $xp = x^_2 = egin{bmatrix} -3 \ -2 \ \ \ \ \ \ \ -1 \ -1 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -4 \ -1 } $ at $t=1$, what is the resulting value of $x^_2$?

<p>-4 (B)</p> Signup and view all the answers

Flashcards

Linear Equation

An equation where each term contains a variable raised to the power of 1.

System of Linear Equations

A set of two or more linear equations with the same variables.

Augmented Matrix

A matrix that represents a system of linear equations; combines the coefficient matrix and constant terms.

Coefficient Matrix

A matrix containing only the coefficients of the variables from a system of linear equations.

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Equivalent Systems

Systems of linear equations that have the same solution set.

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Gaussian Elimination

A method for solving systems of linear equations by performing row operations to transform the augmented matrix into row-echelon form.

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Pivot Variable

The leading variable in a row of the row-echelon form; the variable used to eliminate other variables in subsequent rows.

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Row Operations

Elementary operations performed on the rows of a matrix without changing the solution, including swapping rows, multiplying a row by a non-zero constant, and adding multiple of one row to another.

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Forward Elimination

The first step in Gaussian Elimination where variables are successively eliminated below the pivot variable.

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Degenerate Equation

An equation in a system of equations where the coefficients and constant on the right-hand side all become zero.

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No Solution (System)

A system of linear equations that has no solution; this is indicated in the Gaussian Elimination algorithm when a row with zero coefficients but a non-zero right-hand side constant is reached.

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Row-echelon form

A matrix form after using elementary row operations, where the pivot element in each row appears in a column further to the right than the pivot element in the row above it; creating an upper triangular form.

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Back-Substitution

A technique to solve a system of linear equations in echelon form by substituting values of free variables to find the values of pivot variables.

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Free Variables

Variables in a system of linear equations that can be assigned any value.

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General Solution

A solution to a system of linear equations expressed in terms of free variables, representing all possible solutions.

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Equation Form

Solving a system of linear equations using equations directly.

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Matrix Form

Solving a system of linear equations by using augmented matrices.

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Echelon Matrix

A matrix where, for each non-zero row, the first non-zero element (pivot) is to the right of the pivot in the row above. All zero rows are at the bottom.

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Pivots

Leading non-zero elements in each row of an echelon matrix.

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Row Canonical Form

An echelon matrix with additional properties: each pivot is 1, and all other entries in the pivot's column are 0.

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Unique Solution

A system of linear equations has a unique solution if there is only one set of values for the variables that satisfies all equations.

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Row Equivalence

Two matrices are row equivalent if one can be transformed into the other using a series of row operations.

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What is a state-space model?

A mathematical representation of a dynamical system using state variables, input variables, and output variables. It describes how the system's state changes over time based on inputs and its internal dynamics.

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What is a state vector?

A vector representing the values of all state variables in a system at a specific time. It captures the 'current state' of the system.

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What is an input vector?

A vector representing external signals that affect the system state, such as control signals or external disturbances.

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What is the transition matrix?

A matrix that defines how the state vector changes from one time step to the next based on the system dynamics. It maps the current state to the future state.

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What is the output vector?

A vector representing the system's measurable outputs. What we can observe from the system.

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What is a discrete-time state-space model?

A state-space model where time is divided into discrete intervals, and the state, input, and output are defined at specific points in time.

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State equation

An equation that describes how the state vector evolves over time, based on the current state, input, and system dynamics.

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Output equation

Equation that relates the measurable outputs to the state vector.

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What is the general solution of Ax = b?

The general solution of Ax = b encompasses all possible solutions to the system of linear equations. It is expressed as a sum of a particular solution (xp) and the general solution of the associated homogeneous system (xh).

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What is a particular solution (xp)?

A particular solution of Ax = b is any specific solution that satisfies the system of linear equations. It can be found by setting the free variable(s) to a specific value.

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What is the homogeneous solution (xh)?

The homogeneous solution of Ax = b is the general solution of the associated homogeneous system (Ax = 0). It represents all possible solutions where the right-hand side of the equations is zero.

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How to find the general solution (x*)?

You can obtain the general solution (x*) by finding a particular solution (xp) and then adding the general solution of the homogeneous system (xh). This combines a specific solution with all possible variations that still satisfy the equations.

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What are pivot variables?

Variables in a system of linear equations associated with the leading non-zero coefficients in the row-echelon form of the augmented matrix. These variables are not free to take any value and are dependent on the other variables.

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What are free variables?

Variables in a system of linear equations that are not pivot variables. They can take any value, and their values determine the specific solutions within the general solution.

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What are the steps to solve a system of linear equations?

  1. Express the system as an augmented matrix. 2. Use Gaussian elimination to transform the matrix into row-echelon form. 3. Identify pivot and free variables. 4. Express the general solution in terms of the free variables.
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Why can there be infinite solutions?

A system of linear equations has infinite solutions if there are free variables. These free variables can take on infinitely many values, leading to infinitely many possible solutions.

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Study Notes

Linear Algebra - Chapter 2: Systems of Linear Equations

  • ECE 317: This chapter focuses on systems of linear equations from the textbook "Schaum's Outline of Linear Algebra." It includes supplementary examples from other resources.

Basic Definitions and Solutions

  • Linear Equation: A linear equation in unknowns x₁, x₂, ..., xₙ has the standard form a₁x₁ + a₂x₂ + ... + aₙxₙ = b, where a₁, a₂, ..., aₙ and b are constants.
  • Coefficient of xᵢ: The constant aᵢ.
  • Constant Term: The constant b.
  • Solution: A solution of a linear equation is a list of values for the unknowns (a vector in Kⁿ) that satisfies the equation when substituted.

System of Linear Equations (Systems)

  • Definition 3 (System): A system of linear equations is a set of linear equations with the same unknowns.
  • m × n system: A system with m equations and n unknowns.
  • Square System: A system where the number of equations (m) equals the number of unknowns (n).
  • Homogeneous System: A system where all the constant terms (bᵢ) are zero.
  • Nonhomogeneous System: A system where at least one constant term (bᵢ) is non-zero.
  • Consistent Systems: Systems with at least one solution, potentially unique or infinite.
  • Inconsistent Systems Systems with no solution.

Matrices and Systems

  • Augmented Matrix (M): A matrix that combines the coefficients and constants of a system of equations.
  • Coefficient Matrix (A): The matrix of coefficients. M without the last column.

Degenerate Equations

  • Degenerate Equation: An equation where all coefficients are zero. The solution depends only on the constant 'b'.
  • B≠0 - no solution
  • B=0 infinite set of solutions

Pivot: Leading Variable

  • Pivot: The first nonzero term in a nondegenerate equation. The leading unknown in a non-degenerate equation.

Equivalent Systems and Elementary Operations

  • Equivalent Systems: Two systems have the same solution.
  • Elementary Operations:
  • Interchanging equations: Li ↔ Lj(replace Li with Lj)
  • Multiplying an equation by a nonzero constant: kLį → Lį(Replace Lį with kLį)
  • Adding a multiple of one equation to another: kLi + Lj → Lj (Replace Lj with kLi + Lj)

Small Square Systems (2x2)

  • Geometric Interpretation (2x2): The graph of each equation is a line in a plane (R²).
  • Unique Solution: Distinct slopes.
  • No Solution: Parallel lines, different y-intercepts
  • Infinite Solutions: Coinciding lines.

Systems in Triangular/Echelon Forms

  • Triangular Form: A system where the leading unknown in each subsequent equation appears further to the right,
  • Echelon Form: A system in triangular form or a system with no rows of zeros unless they are below any nonzero rows.

Gaussian Elimination

  • Algorithm 2 (Gaussian Elimination): A method for solving systems of linear equations. • Part A (Forward Elimination): Reduces the system into a simpler, triangular or echelon form using elementary operations. • Part B (Back-Substitution): Solves the simplified system using substitution.

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Explore the concepts of systems of linear equations in this quiz based on Chapter 2 of Schaum's Outline of Linear Algebra. Understand basic definitions, the structure of linear equations, and different types of systems. Test your knowledge with examples and applications presented in this chapter.

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