# Linear Algebra Question:

If you were in Linear Algebra at a graduate level, or have been at some point in the past, what are the main theorems you would think of as being the most important for that course. In the next few days I have an exam and this is very good for my studying. Here's my top 14 list (nice round number) =) , but it might change depending on what other people say:

1. Rank Nullity Theorem

2. Let V and W be finite dimensional vector spaces over F, s.t. dim W = dim V. If T is a linear transformation from V into W, the following are equivalent:
First, T is invertible
Second, T is non-singular
Third, T is onto, i.e. the range of T is W

3. Unique dual basis theorem

4.If f is a polynomial over F with derivative f', then f is a product of distinct irreducible polynomials over F iff f and f' are relatively prime.

5. Let T be a linear operator on a finite dimensional space V and let c be a scalar. TFAE:
First, c is a characteristic/eigenvalue of T.
Second, The operateor (c-TI) is singular, hence not invertible
Third, det(c-TI) = 0

6. Let T be a linear operator on a finite dimensional space V. Let, c1,c2,...,ck be the distinct characteristic/eigenvalues of T and let Wi be the nullspace of (c-TI), TFAE:
First, T is diagonalizable
Second, The characteristic polynomial for T is f = (x-c1)^d1 ... (x-ck)^dk and dim Wi = di, where i = 1,...,k.
Third, dimW1 + dimW2 + ... + dimWk = dimV

7. (Generalized) Cayley-Hamilton Theorem

8. Primary Decomposition Theorem

9.Cyclic Decomposition Theorem

10. Not so much a Theorem, but Gram-Schmidt

11.Inner Product Space Theorem (with properties)

12. For any linear operator T on a finite dimensional inner product space V, there exists a unique linear operator T* on V such that <Ta,b> = <a,T*b>. (T* is called the adjoint of T).

13. Not so much a Theorem, but self-adjoint operators, i.e. Hermitian properties, specifically the fact that there exists an orthonormal basis of eigenvectors and furthermore each eigenvalue of the operator is real.

14. Spectral Theorem