课程名称︰线性代数一
课程性质︰数学系大一必修/经济系选修
课程教师︰杨一帆
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰
考试时限（分钟）：50
试题 :
1. 20pts
For the matrix:
0 1 1
A = ( 2 -1 -2)
-1 1 2
in M_3x3(R)
a) compute its characteristic polynomial and minimal polynomial
b) determine whether A is diagonalizable or not. If A is diagonalizable,
then find an invertible matrix Q s.t. inverse(Q)AQ is diagonal.
2.40 points
Let S and T be two linear operators on a finite-dimensional vector space V.
a) Prove that is S and T commute(i.e., ST=TS), then every eigenspace for T
is an S-invariant subspace of V.
b) Prove that if S is diagonalizable, then the restriction of S to
any nontrivial S-invariant subspace is also diagonalizable.
c)Prove that if S and T are both diagonalizable and ST = TS, then
S and T are simultaneously diagonalizable(that is, there exists a basis
B for V s.t. [S]_B and[T]_B are both diagonal).
d) Prove that if S and T are simulateously diagonalizable, then S and T
commute.
3. 20 points
Let v_1,...,v_k be eigenvectors corresponding to k distinct eigenvalues
λ_1,...,λ_k of a linear operator T on a vector space V. Prove that the
T-cyclic subspace generated by v = v_1 +...+v_k has dimension k.
4. 20 points
Let W_1, ...,W_k be subspaces of a finite-dimensional vector space V s.t.
V = W_1+...+W_k.
Prove that V = W_1⊕...⊕W_k iff dimV = ΣW_j (j=1 to k)