Final Exam Reviewer I 1st Semester Linear Algebra with MATLAB PDF

FINAL EXAM REVIEWER I 1ST SEMESTER I LINEAR ALGEBRA WITH MATLAB 1.1 INTRODUCTION TO Mathematical Representation: SYSTEM OF LINEAR A linear system with n parts can be EQUATION...

FINAL EXAM REVIEWER I 1ST SEMESTER I LINEAR ALGEBRA WITH MATLAB 1.1 INTRODUCTION TO Mathematical Representation: SYSTEM OF LINEAR A linear system with n parts can be EQUATION described using a system of n linear equations in n unknowns. ○ Unknowns: Represent the values of the inputs. ○ Unique Solution: Correct What is a System? analysis ensures a single solution for the input values. A system is composed of several parts that interact and affect one another, producing an output (effect) as a result of an input (cause). Gaussian Elimination: Key Properties of a System: A method for solving a system of linear equations by rewriting the system into 1. Parts Interaction: row-echelon form. Parts of the system influence and interact with each other. Key Process: 2. Input and Output Relationship: The system produces outputs based on 1. Row-Echelon Form: the inputs provided. A simplified form of a matrix where: ○ The first non-zero element in Example: each row (leading entry) is to the right of the one in the row Business Organization above it. ○ Inputs: Capital, employees, raw ○ Rows consisting entirely of materials, factories. zeros are at the bottom. ○ Outputs: Products. 2. Equivalent Systems: ○ Management Role: Determines Rewriting involves creating a series of interactions between inputs to equivalent systems using three basic maximize output. row operations: ○ Swapping two rows. ○ Multiplying a row by a non-zero scalar. Linear Systems ○ Adding or subtracting a multiple of one row to another row. A linear system is one where the 3. Purpose: output is proportional to the input. ○ Simplify the system to isolate Example of Proportionality: variables and solve for them step by step. If all inputs (e.g., capital, employees, raw materials) are doubled, then the Historical Background: output (e.g., production) is also doubled.  FINAL EXAM REVIEWER I 1ST SEMESTER I LINEAR ALGEBRA WITH MATLAB Carl Friedrich Gauss (1777–1855): 1. What is a system? This method is named after the German a) A single part that produces an output mathematician who contributed b) Several parts that interact with each significantly to its development. other to produce an output c) A fixed set of parts with no interaction d) A method of solving equations 2. Which of the following is NOT a key Notes: property of a system? a) Parts interaction Row Operations Do Not Change b) Input and output relationship Solutions: c) Parts act independently Any operations performed maintain d) The system produces outputs based equivalence, ensuring the same on inputs solutions are preserved. 3. What is an example of a linear Gaussian Elimination and system? Applications: a) A system where outputs are not ○ Frequently used in engineering, proportional to inputs physics, and computer science b) A system where doubling all inputs to model and solve real-world doubles the output problems. c) A system that does not involve ○ Particularly useful in solving equations systems with multiple variables d) A system with more outputs than and constraints. inputs Unique Solution Check: 4. In the context of linear systems, what Gaussian elimination ensures a unique does Gaussian elimination aim to do? solution if the system is consistent and a) Find all possible solutions the number of independent equations b) Simplify the system to solve for equals the number of variables. unknowns c) Increase the number of equations d) Eliminate all variables 5. Which of the following is true about Keywords: row operations in Gaussian elimination? System: Input, output, interaction, a) Row operations change the solutions effect, cause. of the system Linear System: Proportionality, input, b) Row operations maintain the output, system of linear equations. equivalence of the system Gaussian Elimination: Row-echelon c) Row operations are used to eliminate form, equivalent systems, row variables permanently operations. d) Row operations have no effect on the Carl Friedrich Gauss: Historical figure system behind Gaussian elimination. 6. What is the historical contribution of Carl Friedrich Gauss to linear systems? a) He developed the concept of TEST YOURSELF: proportionality in systems b) He contributed to the development of FINAL EXAM REVIEWER I 1ST SEMESTER I LINEAR ALGEBRA WITH MATLAB Gaussian elimination c) He created the first linear system d) He solved systems with more variables than equations Linear Equations and Solutions 7. What is a key feature of a system in row-echelon form? A linear equation in n variables a) The first non-zero element in each row is to the left of the one in the row has the general above it form: b) The first non-zero element in each row is to the right of the one in the row above it c) All rows consist of zeros d) There are no leading entries Where: 8. What type of solution does Gaussian elimination guarantee if the system is consistent and the number of coefficients (real independent equations equals the numbers). number of variables? a) No solution constant term (real number). b) Infinite solutions : leading coefficient (associated c) A unique solution d) A contradictory solution with the leading variable. 9. Which of the following operations is NOT part of Gaussian elimination? Solution of a Linear Equation a) Swapping two rows b) Multiplying a row by a non-zero scalar A solution is a set of nnn numbers c) Adding or subtracting a multiple of one row to another that satisfies the d) Dividing one row by a zero scalar equation when substituted. 10. How can you tell if a system of equations has a unique solution when using Gaussian elimination? a) The system must have infinitely many Determining if an Equation is Linear variables b) The system must have more An equation is linear if: equations than unknowns c) The number of independent 1. All variables are raised to the power of equations must equal the number of 1. variables 2. Variables are not multiplied with each d) The system must be inconsistent other. 3. The equation can be written in the form 1.2 SYSTEMS OF LINEAR Examples EQUATION  FINAL EXAM REVIEWER I 1ST SEMESTER I LINEAR ALGEBRA WITH MATLAB For t=1: 1. → Linear Equation 2. → Linear Equation 3. →Non-Linear (exponent on x). FINAL ANSWER: 4. →Non-Linear (product of variables). System of Linear Equations (SLE) 5. → Linear Equation A system of linear equations involves mmm linear equations in nnn unknowns. Parametric Representation of Solution Sets Example: To describe all solutions to a linear equation, a parametric representation is often used: 1. Solve for one variable in terms of the others. 2. Assign a parameter (like t) to the free variables. 3. Represent the solution set parametrically. Types of Solutions Example: 1. Consistent System: ○ At least one solution exists. ○ Examples: 1. Solve for x: Unique solution (one intersection point). Infinite solutions (infinitely many 2. Let y=t, where t is any real number intersection points). (parameter). 2. Inconsistent System: ○ No solution (lines/planes don’t intersect). 3. The parametric solution is: Homogeneous System If , the system is homogeneous. Particular Solution Always has the trivial solution:  FINAL EXAM REVIEWER I 1ST SEMESTER I LINEAR ALGEBRA WITH MATLAB May also have non-trivial solutions (if there are free variables). Notes: 1. Linear vs. Non-Linear Equations: Solving Systems of Linear Equations Check exponents and interactions between variables. Method of Elimination 2. Parametric Solutions: Useful for infinite solutions. 1. Eliminate variables by 3. Homogeneous Systems: Always check adding/subtracting multiples of one for trivial and non-trivial solutions. equation from another. 4. Elimination Method: A systematic 2. Simplify the system into an equivalent, approach ensures accuracy, especially simpler form. for larger systems. Example: Solve the System OTHER EXAMPLES: Example 1: Single Linear Equation Solve parametrically. Solution Steps Solution 1. Eliminate x from the second and third equations: 1. Solve for x: ○ Multiply the first equation by −2: ○ Add to the second equation: 2. Let y=t, where t is a parameter (t ∈ R): 2. Substituting y and z back gives: Solution: 3. Parametric representation of the solution: Final Answer: Particular Solution For t = 0:  FINAL EXAM REVIEWER I 1ST SEMESTER I LINEAR ALGEBRA WITH MATLAB Substitute: ANSWER:: Multiply through by 6: Example 2: System of Linear Equations with Unique Solution Solve the system: 4. Solve for x: Substitute: Solution (Elimination Method) 1. Eliminate y: ○ Multiply the first equation by 3: FINAL ANSWER: ○ Add to the second equation: Example 3: System of Linear 2. Solve for x: Equations with Infinite Solutions From : Solve the system: 3. Substitute into the first equation: Solution (Simplification) 1. Divide the second equation by 2:  FINAL EXAM REVIEWER I 1ST SEMESTER I LINEAR ALGEBRA WITH MATLAB ○ Add to the second equation: The two equations are identical, so they describe the same plane. 2. Parametric solution: Simplified to: y=z ○ Let y=t and z=s, where t, s∈R (free variables). 2. Parametric solution: ○ Solve for x: ○ Let z=t, where t∈R ○ From y=z: y=t ○ Substitute y=t and z=t into Parametric Representation: FINAL ANSWER: For t = 0, s = 0: Parametric Representation:. For t = 1, s = 2: FINAL ANSWER: For t=1: For t=0: Example 4: Homogeneous System of Linear Equations Solve the system: Example 5: System with No Solution Solution Solve the system: 1. Eliminate x: ○ Multiply the first equation by −2: Solution  FINAL EXAM REVIEWER I 1ST SEMESTER I LINEAR ALGEBRA WITH MATLAB 1. Subtract the first equation from the b) The system always has non-trivial second: solutions c) The system always has the trivial solution d) The system has a unique solution Contradiction. 5. Which method is commonly used to solve systems of linear equations? FINAL ANSWER: The system is inconsistent a) Substitution and has no solution. b) Method of Elimination c) Integration d) Graphing 6. What does a system of linear TEST YOURSELF: equations with infinite solutions look like? 1. Which of the following is true about a a) The equations describe the same line linear equation? or plane a) All variables are raised to the power b) The equations describe distinct, of 2 or more parallel lines or planes b) Variables are multiplied with each c) The system has no solutions other d) There is a single intersection point c) All variables are raised to the power 7. In a system with a unique solution, of 1 what can be said about the d) The equation cannot be written in a equations? standard form a) They describe parallel lines or planes 2. Which of the following equations is b) They have no solution NOT linear? c) They intersect at exactly one point or in one line a) d) They describe the same plane 8. How do you recognize a system with b) no solution? a) When the equations describe c) identical lines or planes d) b) When the equations describe parallel 3. What does the parametric lines or planes that do not intersect representation of a solution set c) When the solution is a parametric typically involve? representation a) Solving for one variable in terms of d) When the system has free variables the others 9. What is the trivial solution of a b) Finding the solution by trial and error homogeneous system of linear c) Solving the system without any free equations? variables a) A solution where all variables are d) Assigning random values to the equal to 1 variables b) A solution where all variables are 4. Which of the following describes a equal to 0 homogeneous system of linear c) A solution with infinitely many equations? variables a) The system has no solutions d) A solution with no variables involved  FINAL EXAM REVIEWER I 1ST SEMESTER I LINEAR ALGEBRA WITH MATLAB 10. In the elimination method for solving A matrix is a rectangular array of a system of equations, what is the numbers. It is typically denoted as: first step? a) Multiply one equation by a scalar b) Subtract one equation from the other c) Add multiples of one equation to the other d) Solve one equation for one variable Solving Questions: 1. Solve the system of linear equations: Dimension/Order of Matrix: A matrix with mmm rows and nnn columns is called an m×n matrix. Elementary Row Operations: 2. Solve the system of linear equations: 1. Interchange two rows 2. Multiply a row by a nonzero constant 3. Solve the system of linear equations using elimination: 3. Add a multiple of one row to another row 4. Solve the system of linear equations Row-Echelon Form (REF): using substitution: A matrix is in row-echelon form if: 1. All zero rows are at the bottom. 2. The first nonzero entry in each row 5. Solve the following system of (called leading 1) is 1. equations: 3. For two successive nonzero rows, the leading 1 in the lower row appears further to the right. Reduced Row-Echelon Form (RREF): A matrix in REF is in RREF if: 1.3 GAUSSIAN AND GAUSS 1. Every column containing a leading 1 has JORDAN ELIMINATION zeros in all other positions. Steps for Gaussian Elimination: Matrix  FINAL EXAM REVIEWER I 1ST SEMESTER I LINEAR ALGEBRA WITH MATLAB 1. Write the augmented matrix: Represent the system of linear equations as an augmented matrix. 2. Perform row operations to achieve REF: ○ Use elementary row operations to simplify the matrix Step 2: Perform row operations to step-by-step until it reaches achieve REF. REF. 3. Back-substitution: Convert the REF 1. Eliminate the first entry in the second back into equations and solve for the row (R2​): variables starting from the last row. Steps for Gauss-Jordan Elimination: 1. Write the augmented matrix: Same as Gaussian Elimination. 2. Perform row operations to achieve 2. Eliminate the first entry in the third row RREF: (R3​): ○ Continue simplifying the matrix further from REF until all leading 1s have zeros above and below them. (No change as the first entry in R3​is already 0.) 3. Extract solutions directly from RREF. 3. Eliminate the second entry in R3​ using R2​: Step-by-Step Example: Solve the system using Gaussian Elimination: Step 3: Back-substitution. Solve from bottom to top: Step 1: Write the augmented matrix. ○ From row 3:  FINAL EXAM REVIEWER I 1ST SEMESTER I LINEAR ALGEBRA WITH MATLAB ○ From row 2: FINAL ANSWER: ○ From row 1: Notes: 1. Gaussian Elimination is faster if the solution is only required in REF. FINAL ANSWER: 2. Gauss-Jordan Elimination is more systematic and ideal for solving directly from RREF. 3. Always check for consistency of the system during the process. Using Gauss-Jordan Elimination: TEST YOURSELF: 1. From the REF above, continue to RREF 1. What is the primary difference by clearing entries above and below the between Row-Echelon Form (REF) leading 1s: and Reduced Row-Echelon Form ○ Eliminate the second entry in (RREF)? R1​: a) REF has zeros above the leading 1s; RREF has zeros below them b) REF has zeros below the leading 1s; RREF has zeros above and below them ○ Eliminate the third entry in R2​: c) REF and RREF are the same d) RREF does not have any leading 1s 2. Which of the following is NOT an elementary row operation? a) Interchange two rows ○ Normalize rows by dividing each b) Multiply a row by a non-zero constant row by the leading coefficient. c) Add a multiple of one row to another row d) Multiply a row by zero 3. What is the main purpose of 2. The RREF will yield: back-substitution in Gaussian elimination? a) To achieve Reduced Row-Echelon Form (RREF) b) To find the values of the variables starting from the bottom row c) To eliminate variables from the system  FINAL EXAM REVIEWER I 1ST SEMESTER I LINEAR ALGEBRA WITH MATLAB d) To simplify the matrix to a diagonal 2. Solve the system using form Gauss-Jordan Elimination: 4. In Gauss-Jordan elimination, after reaching Row-Echelon Form (REF), what is the next step? a) Perform back-substitution b) Convert the matrix to Reduced 3. Solve the system of equations using Row-Echelon Form (RREF) by clearing Gaussian Elimination: the entries above and below the leading 1s c) Eliminate all variables d) Solve the system using substitution 5. Which method would you use if you need to find the solution directly from 4. Solve the system using the matrix without back-substitution? Gauss-Jordan Elimination: a) Gaussian elimination b) Gauss-Jordan elimination c) Substitution method d) Graphical method 6. Which of the following is true about the row operations in Gaussian and Gauss-Jordan elimination? a) Row operations only change the 1.4 MATRIX OPERATIONS solution of the system when performed in the wrong order b) Row operations do not affect the solution of the system c) Row operations always change the MATRIX BASICS solution of the system A matrix is a rectangular array of d) Row operations are only used in numbers, symbols, or expressions Gauss-Jordan elimination arranged in rows and columns, enclosed in brackets. Rows: Horizontal lines of entries. Columns: Vertical lines of entries. Solving Questions Dimensions: A matrix with mmm rows and nnn columns is an m×n (read as 1. Solve the following system of "m-by-n") matrix. equations using Gaussian Elimination: Notes: 1. Equality of Matrices: ○ Two matrices A and B are equal (A=B) if: Their dimensions are the same.  FINAL EXAM REVIEWER I 1ST SEMESTER I LINEAR ALGEBRA WITH MATLAB Corresponding entries 1. Addition of Matrices: are equal Two matrices A and B of the same dimension (m×n) can be added: SPECIAL TYPES OF MATRICES Properties: 1. Square Matrix: m=n (rows = columns). Commutative: The main diagonal is the diagonal from Associative: the top left to the bottom right. Identity Matrix for Addition: 2. Diagonal Matrix: Square matrix (n×n) where: Inverse: ○ Only the main diagonal can contain 2. Scalar Multiplication: non-zero values. Multiply every element of a matrix A by 3. Scalar Matrix: a scalar c: Diagonal matrix where all diagonal elements are equal. 4. Identity Matrix: Properties: n×n matrix where: ○ Diagonal elements ○ Non-diagonal elements Example: 3. Matrix Multiplication: A (m×n) can be multiplied by B (n×p) if: ○ Number of columns in A = Number of rows in B. The result AB is an m×p matrix: MATRIX OPERATIONS ​  FINAL EXAM REVIEWER I 1ST SEMESTER I LINEAR ALGEBRA WITH MATLAB Properties: The main diagonal plays a key role in special types of matrices. Associative: For matrix multiplication, order matters: in most cases. When finding the transpose of a product, reverse the order of Distributive: multiplication: Scalar: OTHER EXAMPLES: Matrix Addition: Let: A+B TRANSPOSE OF A MATRIX The transpose of an m×n matrix A Matrix Multiplication: is an n×m matrix where: Let: AB Rows become columns, and columns become rows. Properties: Scalar Multiplication: 1. 1. Let: cA=4 2. 3. 4. SOLUTION: NOTES:  FINAL EXAM REVIEWER I 1ST SEMESTER I LINEAR ALGEBRA WITH MATLAB 2. Let: cB=-2 3. C= SOLUTION: Transpose of a Matrix 1. A= FORMULAS SUMMARY: 1. Addition: 2. B= 2. Scalar Multiplication: 3. Matrix Multiplication: 4. Transpose:  FINAL EXAM REVIEWER I 1ST SEMESTER I LINEAR ALGEBRA WITH MATLAB d) To convert a square matrix into a diagonal matrix 6. Which of the following statements is true about matrix multiplication? a) Matrix multiplication is commutative TEST YOURSELF: b) Matrix multiplication is associative c) The result of matrix multiplication is 1. What is a square matrix? always a square matrix d) Matrix multiplication is always a) A matrix with equal number of rows possible between any two matrices and columns b) A matrix with more rows than columns c) A matrix with more columns than rows Solving Questions d) A matrix that only contains diagonal elements 1. Find: A+B 2. Which of the following is NOT a property of matrix addition? a) Commutative b) Associative c) Distributive over scalar multiplication d) The order of addition affects the result 2. Find: AB 3. Which of the following matrices is a scalar matrix? a) A matrix with non-zero entries only on the main diagonal, and all diagonal elements are equal b) A square matrix where only the main diagonal contains non-zero elements c) A matrix where all entries are equal 3. Find: 3A d) A matrix where all non-diagonal elements are zero 4. When multiplying two matrices A (m×n) and B (n×p), what is the dimension of the resulting matrix AB? 4. Find the transpose of matrix A, a) m×p b) n×m denoted c) p×n d) n×n 5. What is the main purpose of the transpose of a matrix? a) To reverse the elements of the matrix b) To switch rows and columns c) To multiply matrices 5. Find: 2A-B  FINAL EXAM REVIEWER I 1ST SEMESTER I LINEAR ALGEBRA WITH MATLAB A is invertible if and only if (this value is called the determinant of A). 1.4 THE INVERSE OF A MATRIX If , the inverse of A is given by: Matrices and Their Inverses An n×n matrix A is invertible (or nonsingular) if there exists another n×n matrix B such that: Example: Finding the Inverse of a 2×2 Matrix Problem: Find the inverse of where is the identity matrix of order n. The matrix B is called the inverse of A. If a matrix A does not have an inverse, it is called noninvertible (or singular). Step-by-Step Solution: 1. Check if A is invertible: Theorem: Uniqueness of an Inverse Matrix Calculate the determinant ad−bc: If A is an invertible matrix, then its inverse is unique. Since , the matrix A is The inverse of A is denoted by invertible. 2. Apply the formula for the inverse: Substitute Finding the Inverse of a 2×2 Matrix For a matrix into the formula:  FINAL EXAM REVIEWER I 1ST SEMESTER I LINEAR ALGEBRA WITH MATLAB 3. Simplify the scalar multiplication: 1. Determine if the following matrices are invertible. If so, find their inverses: a. Final Answer: b. c. Notes: Identity Matrix ​ The identity matrix of size n×n has 1s on the diagonal and Finding the Inverse of a Matrix Using 0s elsewhere. Gauss-Jordan Elimination The Gauss-Jordan Elimination method involves Example for 2×2: augmenting a square matrix A with the identity matrix III and then performing row operations to transform A into I. The resulting augmented matrix will have I on the left and (the inverse) on the right. Properties of Matrix Inverses: 1. 2. (order matters) Step-by-Step Process 1. Adjoin the Identity Matrix 3. (inverse of the transpose is the transpose of the inverse) Write the augmented matrix , where A is the given matrix and I is the identity matrix of the same order. Practice Problems 2. Row Reduce A to the Identity Matrix Perform elementary row operations (row swapping, scaling, and addition/subtraction of rows) on the  FINAL EXAM REVIEWER I 1ST SEMESTER I LINEAR ALGEBRA WITH MATLAB entire matrix until A on the left becomes the identity matrix I. These operations will simultaneously transform I on the right into. 3. Check if Inverse Exists To be continued.. ○ If A cannot be row-reduced to I, then A is singular and has no inverse. ○ Otherwise, the portion of the matrix that started as I will now be. 4. Verify Your Solution Multiply. If the product is the identity matrix I, then is correct. EXAMPLE #1 3x3 matrix SOLVE THIS PROBLEM: