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Linear Algebra for MATH 308
Lecture 1: Vectors, Linear Independence, and Spanning Sets
Lecture 2: Operations with Matrices and Vectors
Lecture 3: Systems of Equations
Lecture 4: Determinant
Lecture 5: Eigenvectors and Eigenvalues
Lecture 6: Matrix Inverses and Diagonalization
Lecture 7: Systems of Differential Equations
Lecture 8: Systems of Differential Equations
Quantitative Finance
Several Variables Calculus
Section 1: Functions of Several Variables
Section 2: Limits and Continuity
Section 3: Partial Derivatives
Section 4: Tangent Planes and Linear Approximations
Section 5: The Chain Rule
Section 6: Directional Derivatives and the Gradient Vector
Section 7: Maximum and Minimum Values
Section 8: Lagrange Multipliers
Differential Equations
Section 1: Integrating Factor
Section 2: Separable Equations
Section 3: Compound Interest
Section 4: Variation of Parameters
Section 5: Systems of Ordinary Differential Equations
Section 6: Matrices
Section 7: Systems of Equations, Linear Independence, and Eigenvalues & Eigenvectors
Section 8: Homogeneous Linear Systems with Constant Coefficients
Section 9: Complex Eigenvalues
Section 10: Fundamental Matrices
Section 11: Repeated Eigenvalues
Section 12: Nonhomogeneous Linear Systems
Mathematical Probability
Section 1: Probabilistic Models and Probability Laws
Section 2: Conditional Probability, Bayes’ Rule, and Independence
Section 3: Discrete Random Variable, Probability Mass Function, and Cumulative Distribution Function
Section 4: Expectation, Variance, and Continuous Random Variables
Section 5: Discrete Distributions
Section 6: Continuous Distributions
Section 7: Joint Distribution Function, Marginal Probability Mass Function, and Uniform Distribution
Section 8: Independence of Two Random Variables, Covariance, and Correlation
Section 9: Conditional Distribution and Conditional Expectation
Section 10: Moment Generating Function
Section 11: Markov’s Inequality, Chebyshev’s Inequality, and Weak Law of Large Numbers
Section 12: Convergence and the Central Limit Theorem
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Section 6: Matrices
Section 6: Matrices
Instructions
This section covers the concepts listed below.
For each concept, there is a conceptual video explaining it followed by videos working through examples.
When you have finished the material below, you can go to the
next section
or return to the
main Mathematical Probability page
.
Concepts
Matrix Operations
Gaussian Elimination
Inverse of a Matrix
Links & Resources
Return to Differential Equations Page
Return to Mini-Course Main Page
Matrix Operations
Watch Concepts Video
Exercises
Directions:
There are no exercises for this topic.
Self-Assessment Questions
Directions:
The following questions are an assessment of your understanding of the material above. If you are not sure of the answers, you may need to rewatch the videos.
Which matrices can be added?
Define matrix multiplication. Can we multiply any two matrices?
When are both \(AB\) and \(BA\) defined?
Gaussian Elimination
Watch Concepts Video
Exercises
Directions:
There are no exercises for this topic.
Self-Assessment Questions
Directions:
The following questions are an assessment of your understanding of the material above. If you are not sure of the answers, you may need to rewatch the videos.
What are the three elementary row operations?
What is a row echelon form (REF)?
What is the objective of Gaussian elimination?
What is the reduced row echelon form of a matrix?
How do you use Gaussian elimination to see if a given matrix is invertible, and how does it help with finding the inverse?
Inverse of a Matrix
Watch Concepts Video
Exercises
Directions:
You should attempt to solve the problems first and then watch the video to see the solution.
Find the inverse of the matrix \[A=\begin{pmatrix} 1 & -1 & -1 \\ 3 & -1 & 2\\ 2 & 2 & 3\end{pmatrix}\]
Show Answer
\[
\mathbf{A}^{-1}=
\begin{pmatrix}
\frac{7}{10} & -\frac{1}{10} & \frac{3}{10}\\[8pt]
\frac{1}{2} & -\frac{1}{2} & \frac{1}{2}\\[8pt]
-\frac{4}{5} & \frac{2}{5} & -\frac{1}{5}\\[8pt]
\end{pmatrix}
\]
Watch Video Solution
Self-Assessment Questions
Directions:
The following questions are an assessment of your understanding of the material above. If you are not sure of the answers, you may need to rewatch the videos.
Define the inverse of a matrix.
What is the significance of an invertible matrix?
What is the determinant of a matrix, and what is its significance in invertibility?
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