Free online courseLinear Algebra Course

Conteúdo do Curso

• Video class: Linear Algebra 1.1.1 Systems of Linear Equations

  0h18m

• Exercise: In linear algebra, when solving a system of linear equations using an augmented matrix, which operation is NOT an acceptable elementary row operation?

• Video class: Linear Algebra 1.1.2 Solve Systems of Linear Equations in Augmented Matrices Using Row Operations

  0h23m

• Exercise: Which of the following row operations is correctly applied to transform a given matrix?

• Video class: Linear Algebra 1.2.1 Row Reduction and Echelon Forms

  0h17m

• Exercise: What is the key difference between a matrix in echelon form and one in reduced row echelon form?

• Video class: Linear Algebra 1.2.2 Solution Sets and Free Variables

  0h14m

• Exercise: Consider a system of linear equations represented by an augmented matrix in row-echelon form. If you find that the last row of the matrix is [0 0 0 | 5], what does this indicate about the system?

• Video class: Linear Algebra 1.3.1 Vector Equations

  0h12m

• Exercise: Which of the following statements accurately describes a vector in R2?

• Video class: Linear Algebra 1.3.2 Linear Combinations

  0h24m

• Exercise: What is a linear combination of vectors?

• Video class: Linear Algebra 1.4.1 The Matrix Equation Ax=b

  0h11m

• Exercise: Given the matrix A with dimensions 1 by n and a vector X with dimensions n by 1, if A * X results in a vector with all entries zero, what can be said about the vector X?

• Video class: Linear Algebra 1.4.2 Computation of Ax

  0h09m

• Exercise: What is the result of multiplying matrix A = [[2, 1], [-4, 2]] by vector X = [3, -1] using the row vector rule for computing AX?

• Video class: Linear Algebra 1.5.1 Homogeneous System Solutions

  0h17m

• Exercise: What does it mean for a solution to be non-trivial in a homogeneous system of linear equations?

• Video class: Linear Algebra 1.5.2 Non-Homogeneous System Solutions

  0h11m

• Exercise: In the context of linear algebra, what does the general solution to a non-homogeneous system of equations in the form Ax = B typically involve?

• Video class: Linear Algebra 1.6.1 Applications of Linear Systems - Economic Sectors

  0h06m

• Exercise: In a simplified economy with two sectors, goods and services, suppose goods sell 80% of their output to services, and services sell 70% of their output to goods. Using linear systems to find equilibrium, how would the income be matched to expenditures?

• Video class: Linear Algebra 1.6.2 Applications of Linear Systems - Network Flow

  0h08m

• Exercise: In a network flow system, the balance equations for nodes indicate the relationship between incoming and outgoing flows. Given a network where the inflows and outflows are described by the following equations: 1. x1 + x3 = 20 2. x2 - x3 - x4 = 0 3. x1 + x2 = 80 4. x4 = 60 What is the correct expression for the flow pattern for x2 in terms of x3?

• Video class: Linear Algebra 1.7.1 Linear Independence

  0h11m

• Exercise: A set of vectors \( \{\mathbf{v_1}, \mathbf{v_2}, \mathbf{v_3}\} \) is given. Which condition will guarantee that the set is linearly independent?

• Video class: Linear Algebra 1.7.2 Special Ways to Determine Linear Independence

  0h08m

• Exercise: Which of the following statements is true regarding a set of vectors and linear independence?

• Video class: Linear Algebra 1.8.1 Matrix Transformations

  0h13m

• Exercise: In the context of matrix transformations, which of the following statements is true?

• Video class: Linear Algebra 1.8.2 Introduction to Linear Transformations

  0h09m

• Exercise: What is the geometric interpretation of a linear transformation represented by the matrix A = [0 -1; 1 0] when applied to vectors in R²?

• Video class: Linear Algebra 2.1.1 Matrix Operations - Sums and Scalar Multiples

  0h13m

• Exercise: Which of the following describes a square matrix?

• Video class: Linear Algebra 2.1.2 Matrix Operations - Multiplication and Transpose

  0h28m

• Exercise: Given two matrices A and B, with A being a 3x4 matrix and B being a 4x2 matrix, what is the size of the resulting matrix after multiplying A by B?

• Video class: Linear Algebra 2.2.1 The Inverse of a Matrix

  0h14m

• Exercise: Which of the following statements is true regarding the inverse of a 2x2 matrix?

• Video class: Linear Algebra 2.2.2 Solving 2x2 Systems with the Inverse and Inverse Properties

  0h14m

• Exercise: If A is an invertible matrix, what can be said about the inverse of its transpose?