Theorems
线性代数定理作业代写 2 TriangularizationTheorem 2.1. Let V be a finite-dimensional vector space over an algebraically closed field. Every linear transformation T : V
2 Triangularization 线性代数定理作业代写
Theorem 2.1. Let V be a finite-dimensional vector space over an algebraically closed field. Every linear transformation T : V → V is triangularizable.
Theorem 2.2. Let T : V → V be a diagonalizable linear mapping, and let W be a T-invariant subspace of V. The restriction T|W is diagonalizable. Is the same true if “diagonalizable” is replaced by “triangularizable”?
Theorem 2.3. If a,b are (nonzero) algebraic numbers, then a+b, ab, −a, and a−1 are also algebraic. (This shows that Q is a field)
3 Invariant Subspaces 线性代数定理作业代写
Theorem 3.1. A linear mapping P : V → V is a projection if P2 = P.
(a) Let W be any subspace of V. There is a projection P : V → V whose image is W, and that any such projection must satisfy P(w) = w for all w ∈ W.
(b) Let T : V → V be a linear mapping and let P : V → V be a projection with image W = im(P). W is T-invariant if and only if PT P = T P. This shows how invariant subspaces can be interpreted purely in terms of linear mappings: invariant subspaces correspond to commuting with projections.
Theorem 3.2. A linear mapping T :V →V is called decomposable if there are nonzero T-invariant subspaces W1,W2 of V such that V = W1⊕W2. Otherwise, T is called indecomposable.
(a) Show that every diagonalizable mapping is decomposable. Is the converse true? (need to prove)
(b) T is decomposable if and only if there is a basis B of V for which the coordinate matrix of T is “block-diagonal” (need to prove)
(c) Let T : V →V be a linear mapping of a finite-dimensional vector space V. V can be expressed as a direct sum
V = W1⊕··· ⊕Wk
where each Wi is T-invariant, and each restriction T|Wi : Wi → Wi is indecomposable.
4 Jordan Blocks and Cycles 线性代数定理作业代写
Theorem 4.1. Let A ∈ M5(F) be a nilpotent of index 4. Deduce, with proof, the Jordan form of A.
Theorem 4.2. For any linear mapping T,
ker(T2 −I) = ker(T +I)⊕ker(T −I).
5 Minimal Polynomial
Theorem 5.1. Let P : V → V be a linear map. P is a projection if and only if its minimal polynomial divides x2 −x. Can you find a nontrivial projection whose minimal polynomial is not x2 −x? (need to prove)
Theorem 5.2. State the division algorithm for polynomials. Use it to prove that if p(x) ∈ F[x] is a polynomial annihilating a linear mapping T : V → V, then mT (x) divides p(x).
Theorem 5.3. A matrix A has finite order if An = I for some n ∈ N. Every such matrix is diagonalizable. (Hint: does xn −1 have any repeated roots?) (need to prove)
