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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