====== Midterm 2 ======
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)
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