EGN4450 Intro to Linear Systems
EXAM 2 – CH 2 & 3 Study Guide (12
Dr. Jamie Chilton, PhD
U22
th
edition)
EXAM 2 – CH 2 & 3 covers:
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Chapter 2: entire chapter; CH 2 Sections 2.1, 2.2, and 2.3
Chapter 3: first 3 sections only; CH 3 Sections 3.1, 3.2, and 3.3
o Skip rest of Chapter 3 Sections 3.4 – on
20 – 25 Questions Total:
o Multiple Choice and Fill-In-The-Blank
o Problems which require work and scrap paper or calculator
o Simple Proofs similar to Section Examples
o 1 – 2 Challenge Questions to test the application of knowledge
CH 2 Section 2.1: Determinants by Cofactor Expansion
Students should know the following definitions and/or concepts:
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Determinant
Minors and Cofactors
o “Checkboard” array
Cofactor Expansion
Triangular Matrix (upper, lower, and diagonal)
Arrow Technique
Students should be able to:
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Determine if a matrix is invertible
Find the inverse of a matrix
Calculate the determinant of a square matrix
Find the minors and cofactors of a square matrix; use “checkerboard” array
Use cofactor expansion to find the determinant – by row or column
a. Make a smart choice of row or column
6. Calculate the determinant of a square triangular matrix (upper, lower, or diagonal triangular) –
Theorem 2.1.2
7. Use the arrow technique for a 2 x 2 or 3 x 3 square matrix
8. Evaluate the determinant based only on inspection and knowledge of definitions and theorems
CH 2 Section 2.1 problems to review:
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Participation Problems Assigned: CH 2 Section 2.1 Problems 1, 3, 11, 15, and 19
Additional Problems to Review: CH 2 Section 2.1 Problems 21 and 23
EGN4450 Intro to Linear Systems
Solutions to Problems:
Dr. Jamie Chilton, PhD
U22
EGN4450 Intro to Linear Systems
Dr. Jamie Chilton, PhD
U22
EGN4450 Intro to Linear Systems
Dr. Jamie Chilton, PhD
U22
EGN4450 Intro to Linear Systems
Dr. Jamie Chilton, PhD
U22
CH 2 Section 2.2: Evaluating Determinants by Row Reduction
Students should know the following definitions and/or concepts:
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Transpose
Elementary Row Operations
TABLE 1 Relationships and Operations
Elementary Matrix
Proportional Rows or Columns
Row and Column Operations
Students should be able to:
1. Conduct elementary row operations
2. Find the transpose of a matrix
3. Know Theorem 2.2.3 and Table 1 and how elementary row operations affect the value of the
determinant. For instance, bringing a common factor from any row or column of a determinant
through the determinant sign.
4. Evaluate the determinant of elementary matrices
5. Recognize if a matrix has proportional rows or columns and what is the resulting determinant
6. Evaluate determinants by row reduction and Theorem 2.1.2
7. Use column operations and triangular matrices to evaluate the determinant
8. Use combination of row operations and cofactor expansion to evaluate the determinant
9. Evaluate the determinant based only on inspection and knowledge of definitions and theorems
CH 2 Section 2.2 problems to review:
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Participation Problems Assigned: CH 2 Section 2.2 Problems 5, 6, 7, 8, and 9
Additional Problems to Review: CH 2 Section 2.2 Problems 11, 13, and 23
Solutions to Problems:
EGN4450 Intro to Linear Systems
Dr. Jamie Chilton, PhD
U22
EGN4450 Intro to Linear Systems
Dr. Jamie Chilton, PhD
U22
EGN4450 Intro to Linear Systems
Dr. Jamie Chilton, PhD
U22
EGN4450 Intro to Linear Systems
Dr. Jamie Chilton, PhD
U22
CH 2 Section 2.3: Properties of Determinants; Cramer’s Rule
Students should know the following definitions and/or concepts:
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Properties of Determinants
Determinant of Matrix Product
Matrix Invertibility
Determinant Test for Invertibility
Adjoint of a Matrix
Matrix of Cofactors
Inverse Matrix
Cramer’s Rule
Equivalence Theorem
Students should be able to:
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Know and use basic properties of determinants
Evaluate sums of determinants
Evaluate the determinant of the product of square matrices
Evaluate the determinant of the product of a square matrix and an elementary matrix
Determine whether a matrix is invertible by its determinant
Determine the matrix of cofactors from a square matrix
Determine the adjoint of a matrix
Find the inverse of a matrix by its determinant and adjoint
Use Cramer’s Rule to solve a linear system
CH 2 Section 2.3 problems to review:
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Participation Problems Assigned: CH 2 Section 2.3: none assigned
Additional Problems to Review: CH 2 Section 2.3 Problems 5, 10, 11, 15, 19, and 25
EGN4450 Intro to Linear Systems
Solutions to Problems:
Dr. Jamie Chilton, PhD
U22
EGN4450 Intro to Linear Systems
Dr. Jamie Chilton, PhD
U22
CH 3 Section 3.1: Vectors in 2-Space, 3-Space, and n-Space
Students should know the following definitions and/or concepts:
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Geometric Vectors
o Direction
o Length/Magnitude
o Initial and Terminal Points
o Equivalent
Zero Vector
Vector Addition – Parallelogram Rule and Triangle Rule
o Translation
Vector Subtraction
o Negative
o Difference
Scalar Multiplication
o Scalars
o Scalar Product
Parallel and Collinear Vectors
Associative Law for Addition
Ordered n-tuples, Components, and Coordinates
Real n-space Rn
Equality
Algebraic Operations using Components
Linear Combinations with Coefficients and Vectors
Vectors in comma-delimited, row-vector, and column-vector forms
Students should be able to:
EGN4450 Intro to Linear Systems
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Dr. Jamie Chilton, PhD
U22
Recognize the initial and terminal point of a vector
Determine if vectors are equivalent
Add vectors
Subtract vectors
Multiply vectors by a scalar
Determine if vectors are parallel
Find the components of a vector, especially if the initial point is not at the origin
Find the initial and terminal points of vectors
Determine if vectors are equal
Express a vector as a linear combination of other vectors
Express vectors in comma-delimited, row vector, and column vector form
CH 3 Section 3.1 problems to review:
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Participation Problems Assigned: CH 3 Section 3.1 Problems 1, 2, 3, 5, 7, 10, 19, 20, and 23
Solutions to Problems:
EGN4450 Intro to Linear Systems
Dr. Jamie Chilton, PhD
U22
EGN4450 Intro to Linear Systems
Dr. Jamie Chilton, PhD
U22
CH 3 Section 3.2: Norm, Dot Product, and Distance in Rn
Students should know the following definitions and/or concepts:
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Pythagoras Theorem
Norm of a Vector (length, magnitude) and its notation
Zero Vector
Unit Vectors and Standard Unit Vectors
Normalizing a Vector
Distance in Rn
Dot Product (Euclidean Inner Product) using norms and cos θ
Angle θ between Vectors (acute, obtuse, right)
Component Form of Dot Product
Algebraic Properties of the Dot Product
Cauchy-Schwarz Inequality
Triangle inequality for vectors
Triangle inequality for distance
Parallelogram Equation for Vectors
Dot Products as Matrix Multiplication: TABLE 1
Students should be able to:
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Recognize the notation for the norm of a vector
Find the norm of a vector
Find unit vectors in the same or opposite direction of a given vector
Express linear combinations of standard unit vectors
Calculate the distance between vectors
Recognize the notation for the dot product
Evaluate the dot product of nonzero vectors using norms and cos θ
Find the angle θ between vectors
Determine if the angle θ between vectors is acute, obtuse, or right
Evaluate the dot product of nonzero vectors using their components
Find the norm of a vector in terms of a dot product
Use the algebraic properties of dot products and determine if expressions make sense
mathematically
CH 3 Section 3.2 problems to review:
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Participation Problems Assigned: Problems 2, 4
Additional Problems to Review: CH 3 Section 3.2 Problems 8, 10, 12
EGN4450 Intro to Linear Systems
Solution to Problems:
Dr. Jamie Chilton, PhD
U22
EGN4450 Intro to Linear Systems
Dr. Jamie Chilton, PhD
U22
CH 3 Section 3.3: Orthogonality
Students should know the following definitions and/or concepts:
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Orthogonal Vectors (perpendicular)
Zero Vector
Normal, n
Point-normal equation of line or plane
Component equation of line or plane
Vector form of a line or plane
Orthogonal Projections (“decompose” a vector)
o Vector component of u along a
o Vector component of u orthogonal to a
Norm of a Projection
Theorem of Pythagoras in Rn
Students should be able to:
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Determine if vectors are orthogonal (perpendicular)
Given an equation for a line or plane, determine its points and normal
Find the point-normal form of the equation for a line or plane given a point and a normal
Determine whether a line or plane goes through the origin
Recognize homogenous equations
Determine if lines or planes are parallel or perpendicular
Recognize the notation for orthogonal projections
Evaluate orthogonal projections of a vector
a. Find the vector component of u along a
b. Find the vector component of u orthogonal to a
9. Recognize the notation for the norm of a projection
10. Calculate the norm of a projection
11. Use the Theorem of Pythagoras in Rn
CH 3 Section 3.3 problems to review:
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Participation Problems Assigned: CH 3 Section 3.3 Problems 2, 4, 8, 12, 14, and 18
Solutions to Problems:
EGN4450 Intro to Linear Systems
SKIP CH 3 SECTIONS 3.4 & 3.5!
Dr. Jamie Chilton, PhD
U22
EGN4450 Intro to Linear Systems
Dr. Jamie Chilton, PhD
U22
CH 2 & 3 – DEFINITIONS & THEOREMS (12
th
edition)
2.3 Properties of Determinants; Cramer’s Rule
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Suppose that A and B are n x n matrices and k is any scalar, then
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In general, det(A + B) will usually not be equal to det(A) + det(B), except in special cases
(Theorem 2.3.1).
𝐝𝐝𝐝𝐝𝐝𝐝(𝑨𝑨 + 𝑩𝑩) ≠ 𝒅𝒅𝒅𝒅𝒅𝒅 (𝑨𝑨) + 𝒅𝒅𝒅𝒅𝒅𝒅 (𝑩𝑩)
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But in special cases, see Theorem 2.3.1:
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Determinant of a Matrix Product
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Determinant Test for Invertibility
EGN4450 Intro to Linear Systems
Dr. Jamie Chilton, PhD
U22
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Adjoint of a Matrix: If the entries in any row are multiplied by the corresponding cofactors from
a different row, the sum of these products is always zero.
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Cramer’s Rule: Formula for the solution of a linear system Ax = b of n equations in n unknowns
in the case where the coefficient matrix A is invertible (or, equivalently, when det(A) ≠ 0).
EGN4450 Intro to Linear Systems
Dr. Jamie Chilton, PhD
U22
EGN4450 Intro to Linear Systems
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Dr. Jamie Chilton, PhD
U22
EQUIVALENCE THEOREM that relates all of the major topics we have studied thus far: