MATH 211 • Midterm II • Linear Algebra

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

Exam-like Question 4

Span of Vectors

Span of Vectors

Span of Vectors

Span of Vectors

Span of Vectors

Linear Independence

Linear Independence

Linear Independence

Linear Independence

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

Null Space & Column Space & Row Space

Bases for Col A & Row A & Null A

Exam like Question

Basis for Vector Subspaces

Basis for Matrix Subspaces

Basis for Polynomial Subspaces

Basis for Polynomial Subspaces 2

Rank and Nullity - Theoretical

Null & Column & Row Spaces 1

Null & Column & Row Spaces 2

Null & Column & Row Spaces 3

Null & Column & Row Spaces 4

Null & Column & Row Spaces 5

Column,Null and Row Spaces 1

Column,Null and Row Spaces 2

Column,Null and Row Spaces 3

Basis for Subspaces (Vectors)

Basis for Subspaces (Vectors)

Basis for Subspaces (Vectors)

Basis for Subspaces (Vectors)

Basis for Subspaces (Matrices)

Basis for Subspaces (Matrices)

Basis for Subspaces (Matrices)

Basis for Subspaces (Matrices)

Basis for Subspaces (Matrices)

Basis for Subspaces (Matrices)

Basis for Subspaces (Matrices)

Basis for Subspaces (Matrices)

Basis for Subspaces (Polynomials)

Basis for Subspaces (Polynomials)

Basis for Subspaces (Polynomials)

Basis for Subspaces (Polynomials)

Spanning Sets (Vectors)

Spanning Sets (Vectors)

Spanning Sets (Vectors)

Spanning Sets (Vectors)

Spanning Sets (Polynomials)

Spanning Sets (Polynomials)

Spanning Sets (Polynomials)

Spanning Sets (Matrices)

Linear Independence of Polynomials

Coordinate Systems : Introduction

Coordinate Systems : Polynomials

Exam-like Question 1

Exam-like Question 2

Exam-like Question 3

Exam like Question 4

Exam like Question 5

Key concepts for orthogonality

Orthogonal Complement

Orthagonal Complement 2

Exam-like Question 1

Exam-like Question 2

Exam-like Question 3

Exam like Question 4

Orthogonal Sets and Basis

Advantage of an Orthogonal Basis

Orthonormal Sets and Basis

Orthogonal Matrix

Basis for Orthogonal Complement

Orthogonal Projection 1

Orthogonal Projection 2

Exam-like Question 1

Exam-like Question 2

Exam-like Question 3

The Gram-Schmidt Process 1

The Gram-Schmidt Process 2

Exam like Question 1

Exam like Question 2

Angle Between Vectors for a Given Inner Product

Linear Transformations 1

Linear Transformation 2

Matrix of a Linear Transformation

Inverse of Linear Transformation 1

Inverse of Linear Transformation 2

Linearity of Transformations

Exam like Question 1

Exam like Question 2

Exam like Question 3

Exam like Question 4

Basis for Kernel and Range of a Transformation 1

Basis for Kernel and Range of a Transformation 2

Basis for Kernel and Range of a Transformation 3

Linear Independence of Matrices

Basis for Kernel and Range of a Transformation 4

Basis for Kernel and Range of a Transformation 5

Basis for Kernel and Range of a Transformation 6

Transformation Matrix Relative to Basis 1

Transformation Matrix Relative to Basis 2

Transformation Matrix Relative to Basis 3

Exam like Question 1

Exam like Question 2

Eigenvalues & Eigenvectors : 3 Vital Steps

Eigenvalues & Eigenvectors: A Special Tip

A Matrix with Irrational Eigenvalues

Algebraic and Geometric Multiplicities 1

Algebraic and Geometric Multiplicities 2

Eigenspace of Matrices

Diagonalization 1

Diagonalization 2

Exam like Question 1

Exam like Question 2

Linear Transformation 1

Linear Transformation 2

Matrix of Linear Transformation 1

Matrix of Linear Transformation 2

Inverse of Linear Transformation 1

Inverse of Linear Transformation 2

Inverse of Linear Transformation 3

Composite of Linear Transformation 1

Reflection of a Transformation 1

Reflection of a Transformation 2

Reflection of a Transformation 3

Basis for Kernel and Range 1

Basis for Kernel and Range 2

Basis for Kernel and Range 3

Basis for Kernel and Range 4

Basis for Kernel and Range 5

Basis for Kernel and Range 6

Transformations & Change of Basis 1

Transformations & Change of Basis 2

Transformations & Change of Basis 3

Transformations & Change of Basis 4

Transformations & Change of Basis 5

Transformations & Change of Basis 6

Transformations & Change of Basis 7

Transformations & Change of Basis 8

Eigenvalues

Eigenvalue Eigenvector Diagonalizable

Diagonalization 1

Diagonalization 2

Diagonalization 3

Diagonalization 4

Diagonalization 5

Transformations & Eigenvalues 1

Transformations & Eigenvalues 2

Transformations & Eigenvalues 3

Transformations & Eigenvalues 4

Eigenvalue and Null Space

Determinant

Basis and Coordinate

Rank, Null and Column Space

Subspace, Rank and Null Space

Orthogonal Complement

Linear Transformation

Vector Projection

Orthogonal Vectors

Orthogonal Complement

Diagonalization