This course will need you to be fluent in basic linear algebra and calculus. Please check if you can answer the following questions:
Linear algebra
- How do you do matrix multiplication?
- How do you calculate the eigenvalues and eigenvectors of a square matrix?
- What is the definition of a positive definite/positive semidefinite/negative definite/ negative semidefinite/indefinite matrix?
- What is the rank of a square matrix?
- What is the property of a singular matrix?
- What is a symmetric matrix?
- Let \({\bf X}\) be a \(n\)-by-\(p\) matrix, what can you tell about \({\bf X}^T{\bf X}\)?
Calculus
- What is the definition of a gradient?
- What is the geometric meaning of a gradient?
- What is a contour line?
- What is the relationship between a contour line and a gradient on that line?
- Let a function be \(f(x_1,x_2,...,x_p) = a_1x_1+a_2x_2+...+a_px_p\). Write the matrix form of this function.
- Let \(f({\bf x}) = {\bf A}{\bf x}\) where \({\bf A}\) is \(n\)-by-\(p\) and \({\bf x}\) is \(p\)-by-\(1\). Calculate the gradient \(\frac{\partial f}{\partial {\bf x}}\).
- Let \(f({\bf x}) = {\bf x}^T{\bf A}{\bf x}\) where \({\bf A}\) is \(p\)-by-\(p\) and \({\bf x}\) is \(p\)-by-\(1\). Calculate the gradient \(\frac{\partial f}{\partial {\bf x}}\).
- What is the definition of a Hessian matrix?
- What is the property of a Hessian?
- Perform 2nd order Taylor’s expansion on this function with respect to \(x=0\): \(f(x) = \exp(ax)\) where \(a\) is a scalar.
- Perform 2nd order Taylor’s expansion on this function with respect to \({\bf x}={\bf 0}\): \(f({\bf x}) = \exp({\bf a}^T{\bf x})\) where \({\bf a}\) and \({\bf x}\) are \(p\)-by-\(1\) vectors.
- What is a convex set?
- What is a convex function?
Tutorials
If you have difficulty answering these questions, please check out the following tutorial:
Password is: ME555
These tutorials are prepared by Emrah Bayrak and others at the Optimal Design Lab, the University of Michigan, and only shared with this class. Please DO NOT share the videos outside of the class.