We will cover Chapter 3 and Chapter 4.1, eigenvalue problem.

Key concepts and skills

• Given a linear map $T: V \to V'$, what are $\ker(T), im(T)$, what is $rank(T)$?
• If $W \In V$ is a subspace, then what's the relation between dimension of $W$, $V/W$ and $V$?
• Given a linear map $T: V \to V$, where $V$ is of dimension $n$, does it make sense to talk about $\det(T)$? Why? Given a linear map $T: V \to W$, both $V,W$ are of dimension $n$, does it make sense to talk about $\det(T)$? Why?
• What is Gauss Elimination? The input is what? And what do we want to achieve?
• Given a system of linear equations, $A x = b$, abstractly, how do we know if there are solutions? if there are unique solutions? How to describe the 'non-uniqueness' of the solution?
• Concretely, if one is given $A x = b$ with numbers, how to solve it?
• What is LPU decomposition?
• What is a complete flag in $K^n$? How does a linear transformation acts on a flag?
• What's Inertia theorem about? (find a basis so that a quadratic form looks simple).
• What is the difference between classifying a complex symmetric form, and a real symmetric form?
• What is the difference between a Hermitian sesquilinear form and a symmetric bilinear form?
• What does 'diagonalization' mean? (find a nice basis, so the matrix form of whatever object one look at is a diagonal matrix)
• Eigenvalues: when can you diagonalize, and when you cannot diagonalize? (check matrix to see if it is normal)