110年 - 110 國立臺灣大學_碩士班招生考試_應用數學科學研究所應用數學組:線性代數(A)#100602

(a) Prove that rank(A+B)≤ rank(A)+ rank(B).

2. (15 points.) Let A be an n x n matrix over C of the form 

DefineProve that

3. (15 points.) Let T : V→V be a linear operator on a finite-dimensional vector space be a polynomial. Prove that the linear transformation f (T) is invertible if and only if f(z) and the minimal polynomial T have no common roots.

4. (15 points.) Letbe eigenvectors corresponding to I: distinct cigenvalues of a lincar operator T on a vector space V. Prove that the T-cyclic subspace generated by has dimension k.

5. (15 points.) Let T : V →V be a lincar operator on a finite-dimensional inner product space V over R and T* be its adjoint. Suppose that T* = T³. Prove that T² is diagonalizable over R.

6. (20 points.) Let V be a v vector space of dimension n over a field F. Determine the dimension over F of the vector space of multilinear alternating functions f : V x... xV →F(k copies of V).