Math 334 theorems, definitions

Vector space

Must have vector addition and scalar multiplication that satisfy: commutativity, associativity, additive identity, additive inverses, multiplicative identity, distribution

Subspace Criterion

Let V be a vector space and U a subset of V. U is a subspace of V iff: 1) U contains zero vector aka additive identity, 2) U is closed under vector addition, 3) U is closed under scalar multiplication

Condition for a Direct Sum

Let V be a vector space, and let U1, U2, ..., Um be subspaces of V. Then U1 + ... + Um is direct iff whenever we write 0 = u1 + ... + um for uj in Uj, we have u1 = ... = um = 0

List Length Thm

If V has n dimensions, a list of linearly independent vectors in V has length <= n

Constructing Basis Thm

Let V be a finite dimensional vector space: 1) V has a basis, 2) Every spanning list for V contains a basis, 3) Every linearly independent list for V extends to a basis

Constructing Linear Maps Thm

Let V, W be vector spaces over field F, and assume V is finite-dimensional. Let v1, ..., vn be a basis for V (dim V = n). Let w1, ..., wn be a list in W of length n. There exists a unique linear transformation T from V to W with Tvj = wj for each j.

Rank Nullity Thm

Let V and W be vector spaces over a field F and assume that V is finite-dimensional. If T is a linear transformation from V to W, then null(T) and range(T) are finite-dimensional. Also, dim V = rank(T) + nullity(T)

Incompatible Dimensions Thm

Let V and W be vector spaces over a field F. Assume that V is finite-dimensional, and that T is a linear transformation from V to W. 1) If W is finite-dimensional and dim W < dim V, then T is not injective. 2) if W is finite-dimensional and dim W > dim V (or if W is infinite-dimensional) then T is not surjective

Algebra with Linear Maps Thm

Let V, W, U, E be vector spaces over a field F. 1) The set of linear maps from V to W is a vector space over F. 2) Associativity, distribution, and identity

Compatible Dimensions Thm

Let V, W be finite-dimensional vector spaces over F.

1) dim V = dim W iff V, W isomorphic

2) Let T be a linear transformation from V to W. If dim V = dim W then the following are equivalent:

T injective, T surjective, T invertible. They are either all true or all false.

Linear Maps and Subspaces Thm

Let V, W be vector spaces over F and T a linear transformation V -> W.

1) For every subspace U of V, T(U) is a subspace of V.

2) For every subspace E of W, T^{-1}(E) is a subspace of V.

Star-jective Linear Maps Thm

Let V, W be vector spaces over F. Let T a linear transformation V -> W

1) T is surjective iff range(T) = W

2) T is injective iff null(T) is {0}

3) T is bijective iff there exist a linear transformation S from W -> V such that TS = Iw and ST = Iv

Dimension Theorem

Let V be a finite-dimensional vector space. Then there exists natural number n such that:

1) Every basis for V has length n

2) Every spanning list for V of length n is a basis

3) Every linearly independent list in V of length n spans V

Polynomials and Invariance Thm

Let V be a vector space, T an operator on V, and p, q are polynomials over a field. Then null p(T) and range p(T) are invariant under q(T)

Conditions for Eigenvalues Thm

Let V be a finite-dimensional vector space, T an operator on V, and lambda in the field. TFAE:

1) lambda is an eigenvalue of T

2) lambda is in the spectrum of T

3) T - lambda(I) is not injective

4) T - lambda(I) is not surjective

Linearly Independent Eigenvectors Thm

Let V be a vector space and T an operator on V. Let lambda_1, …, lambda_m be a list of eigenvalues of T. Assume v_j is an eigenvector of T corresponding to an eigenvalue lambda_j. Then eigenvectors corresponding to distinct eigenvalues are linearly independent.

Existence + Uniqueness of Minimal Polynomial Thm

Let V be a finite-dimensional vector space and T an operator on V. There is a unique monic polynomial of smallest degree such that p(T) = zero map. Also, if V is nontrivial, degree of p >= 1.

Minimal Polynomial divides polynomials that destroy T

Let V be a finite-dimensional vector space, let T be an operator on V, let f be a polynomial over a field. Then f(T) = 0 iff there is a polynomial q such that f = (minimal polynomial)*(q); the minimal polynomial divides f.

Eigenvalues and Minimal Polynomials Thm

Let V be a finite-dimensional vector space, let T be an operator on V, let lambda be a scalar in a field. Then (min poly)(lambda) = 0 iff lambda is an eigenvalue of T. Basically the roots of the minimal polynomial are the eigenvalues of T.

Fundamental Thm of Algebra + Factoring Polynomials over R

Let p be a polynomial over a field with deg p >= 1.

1) If p is over the complex field, p can be factored into linear terms.

2) If the field is the real numbers, then p can be factored into quadratic and linear terms.

Eigenvalues and Matrices Thm

Let V be a fin-dim vector space with dim(V) = n, and T an operator on V. Let B: v1, …, vn be a basis for V, and A is the matrix with respect to B of T.

1) A is diagonal iff each diagonal entry, A_k,k is an eigenvalue of T where v_k is an eigenvector of T corresponding to eigenvalue A_k,k

2) If A is upper-triangular then:

i) T is invertible iff 0 is not an eigenvalue

ii) Spectrum of T is exactly the diagonal entries of A

Basis Conditions for Upper-Triangular Matrices

Let V be a finite-dimensional vector space with basis B: v_1, …, v_n, and let T be an operator on V. Let A be the matrix with respect to B of T. Then TFAE:

1) A is upper-triangular

2) For each 1 <= k <= n, Tv_k is in span(v_1, …, v_k)

3) span(v_1, …, v_k) is invariant under T for each 1 <= k <= n

Moreover, if any (and therefore all) of these conditions hold, then T satisfies:

(T - A_1,1I)…(T - A_n,nI) = 0 → minimal polynomial divides the characteristic polynomial

Polynomial Condition for Upper-Triangular Matrix

Let V be a finite-dimensional vector space, and let T be an operator on V. Then TFAE:

1) There exists a basis B for V such that the matrix with respect to B of T is upper-triangular

2) There is a list lambda_1, …, lambda_m of scalars satisfying p_T(x) = (x - lambda_1)…(x - lambda_m) → this list can have repetitions

Eigenspace Lemma

Let V be a vector space and T an operator on V. Let lambda in field and v a vector in V.

1) E(lambda, T) is a subspace of V and is invariant under T (eigenspace of lambda, T)

2) T applied to any vector in E(lambda, T) = lambda*identity

3) lambda is an eigenvalue of T iff E(lambda, T) ≠ {0}

4) v is an eigenvector of T corresponding to eigenvalue lambda of T iff v in E(lambda,T) and v nonzero

5) If V is finite-dimensional and lambda_1, …, lambda_m are distinct eigenvalues of T, then E(lambda_1, T) + … + E(lambda_m, T) is direct and dim E(lambda_1, T) + … + dim E(lambda_m, T) <= dim V

Diagonalizability Thm

Let V be a finite-dimensional vector space with dim V = n, let T an operator on V, and write spectrum(T) = {lambda_1, …, lambda_m} where each lambda is distinct. Then TFAE:

1) T is diagonalizable

2) There exists a basis B: v_1, …, v_n for V where each v is an eigenvector of T

3) V = direct sum of all eigenspaces

4) dim V = sum of dims of eigenspaces

Polynomial Condition for Diagonalizability Thm

T is diagonalizable iff p_T(x) = (x - lambda_1)…(x - lambda_m) for some distinct lambda_1, …, lambda_m in F. Basically the roots of the minimal polynomial are exactly the eigenvalues of T.