Linear Algebra Theorems for Graduate Level Course

[CoachZ]

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

 

[brian44]

I would also put Rank + Nullity Theorem at the top, or the RCF theorem (see below).

Perhaps we have been following a bit different set of topics than in your course, but for us one of the most important theorems was the Rational Canonical Form Theorem, and subsequently Jordan Canonical Form - so many things after derive from these. (Maybe this is what you refer to as primary and cyclic decomposition theorems). This gave us way to represent any linear operator on a finite dimensional vector space using its minimal polynomial by a block-diagonal matrix with each block corresponding to a term of the minimal polynomial, and blocks of companion matrices along the diagonal of each primary block.

Otherwise I would agree with your list, I can't think of much else. One other thing may be change-of-basis matrix and similarity, we say A is similar to B if A = PBP^-1 for some invertible P, a change of basis matrix, though this is related to the discussion of canonical form theorems.