MATH 225 • Midterm • Linear Algebra and Differential Equations

Augmented Matrix - Row Operations 1

Augmented Matrix - Row Operations 2

Existence and Uniqueness of Solution

Free Variables

Exam-like Questions 1

Exam-like Questions 2

Exam like Question 3

Fundamental Matrix Operations

Inverse of a Matrix 1

Inverse of a Matrix 2

Exam-like Question 1

Exam-like Question 2

Exam like Question 3

Matrix Equation

Solving AX = B with Inverse

Exam like Question 1

Exam-like Question 2

Exam like Question 3

Elementary Matrix

Inverse of Elementary Matrix

Exam-like Question 1

Exam-like Question 2

Exam-like Question 3

Determinant & Cofactor Expansion

Determinants of Bigger Matrices

Properties of Determinant

Properties of Determinant 2

Cramer's Rule

Adjoint Matrix

Exam-like Question 1

Exam-like Question 2

Exam-like Question 3

Vector Properties

Linear Combination 1

Linear Combination 2

Formal Definition of Linear Combination

Span of Vectors 1

Span of Vectors 2

Solutions of Homogeneous Linear Systems

Exam-like Question 1

Exam-like Question 2

Exam-like Question 3

Linear Independent Sets

Linear Independence of Matrices

Special Theorem for Matrix Equation

Exam-like Questions 1

Exam-like Questions 2

Exam-like Question 3

Vector Spaces and Subspaces

Polynomial Subspaces

Matrix Spaces & Matrix Subspaces

Exam-like Question 1

Exam-like Question 2

Exam-like Question 3

Exam-like Question 4

Exam-like Question 5

Exam-like Question 6

Exam-like Question 7

Null Space & Column Space & Row Space

Bases for Col A & Row A & Null A

Basis for Vector Subspaces

Basis for Matrix Subspaces

Basis for Polynomial Subspaces

Basis for Polynomial Subspaces 2

Exam-like Question 1

Exam-like Question 2

Exam-like Question 3

Exam-like Question 4

Key concepts for orthogonality

Orthogonal Complement

Orthagonal Complement 2

Orthogonal Sets and Basis

Exam-like Question 1

Exam-like Question 2

Exam-like Question 3

Eigenvalues & Eigenvectors : 3 Vital Steps

Eigenvalues & Eigenvectors: A Special Tip

A Matrix with Irrational Eigenvalues

Algebraic and Geometric Multiplicities

Eigenspace of Matrices

Complex Eigenvalues

Complex Eigenvalues 2

Diagonalization 1

Diagonalization 2

Exam like Question 1

Exam like Question 2

Existence and Uniqueness 1

Existence and Uniqueness 2

Existence and Uniqueness 3 (Fall 2021)

Matrix Operations (Spring 2022)

Elementary Matrices 1

Elementary Matrices 2 (Fall 2021)

Determinant and Invertibility

Properties of Determinants 1

Properties of Determinants 2 (Fall 2020)

Subspace or not 1

Subspace or not 2

Subspaces or not 3 (Fall 2022)

Subspaces or not 4 (Fall 2022)

Subspaces or not 5

Basis for Subspaces 1

Basis for Subspaces 2 (Spring 2023)

Basis for Subspaces 2

Basis for Subspaces 3

Basis for Subspaces 4

Basis for Subspaces 5 (Spring 2022)

Polynomials Basis 1

Polynomials Basis 2

Polynomials Basis 3

Polynomials Basis 4 (Fall 2020)

Polynomials Basis 5

Null & Column & Row Spaces

Null & Column & Row Spaces 2

Null & Column & Row Spaces 3 (Fall 2020)

Null & Column & Row Spaces 4

Null & Column & Row Spaces 5 (Fall 2021)

Diagonalization 1

Diagonalization 2

Orthogonal Complement 1

Orthogonal Complement 2

Orthogonal Complement 3

Orthogonal Complement 4

Orthogonal Complement (Fall 2021)

Orthogonal Complement (Spring 2023)

Orthogonal Complement (Fall 2020)

Theory Problem 1

Theory Problem 2

Theory Problem 3 (Spring 2022)

Theory Problem 4 (Spring 2022)

Existence and Uniqueness of Solution

Determinants

Elementary Matrices

Determinant and Invertibility

Basis and Subspaces

Basis and Subspaces