108B HW 2
YOUR NAME HERE
April 10, 2021
Lecture 3
3D
Problem 4. Suppose W is finite-dimensional and T1, T2 ∈ L(V,W ). Prove
that KerT1 = KerT2 if and only if there exists an invertible linear operator
S ∈ L(W ) such that T1 = ST2.
Solution.
1
Problem 5. Suppose V is finite-dimensional and T1, T2 ∈ L(V,W ). Prove
that rangeT1 = rangeT2 if and only if there exists an invertible linear operator
S ∈ L(V ) such that T1 = T2S.
Solution.
2
Problem 6. Suppose V and W are finite-dimensional and T1, T2 ∈ L(V,W ).
Prove that there exist invertible operators R ∈ L(V ) and S ∈ L(W ) such that
T1 = ST2R if and only if dim KerT1 = dim KerT2.
Solution.
3
5A
Problem 15. Suppose T ∈ L(V ), and S ∈ L(V ) is invertible.
(a) Prove that S and S−1TS have the same eigenvalues.
(b) What is the relationship between the eigenvectors of T and the eigenvec-
tors of S−1TS?
Solution.
4
5C
Problem 3. Suppose V is finite-dimensional and T ∈ L(V ). Prove that the
following are equivalent:
(a) V = KerT ⊕ rangeT .
(b) V = KerT + rangeT .
(c) KerT ∩ rangeT = {0}.
5
Lecture 4
5C
Problem 5. Suppose V is a finite-dimensional complex vector space and T ∈
L(V ). Prove that T is diagonalizable if and only if
V = Ker(T − λI)⊕ range(T − λI)
for every λ ∈ C.
Solution.
6
Problem 10. Suppose that V is finite-dimensional and T ∈ L(V ). Let λ1, . . . , λm
denote the distinct nonzero eigenvalues of T . Prove that
m∑
i=1
dimE(λi, T ) ≤ dim rangeT.
Solution.
7
5B
Problem 2. Suppose T ∈ L(V ) and (T − 2I)(T − 3I)(T − 4I) = 0. Suppose λ
is an eigenvalue of T . Prove that λ ∈ {2, 3, 4}.
Solution.
8
Problem 4. Suppose P ∈ L(V ) and P 2 = P . Prove that V = KerP⊕rangeP .
Solution.
9
Problem 20. Suppose V is a finite-dimensional complex vector space and T ∈
L(V ). Prove that T has an invariant subspace of dimension k for each k ∈
{1, . . . ,dimV }.
Solution.
10

欢迎咨询51作业君