## Transcribed Text

Problem 1. Ture or False? (To practice, if false, please modify each wrong statement to
correct one. or give some counterexample)
(1) For an non invertible n 72 matrix A, The mank of A plus the algebraic multiplicity of
eigenvalue O ia equal to the size of A
(2) A diagonalizable if and only if the geometric multiplicity and algebraic multiplicity of each
eigenvalue are equal.
(3) if / a 2 by matrir and frace of A is 0. then the determinant of 4 is not able to be
greater than
(4) For any matriz M. we can find some invertible P such that PM is the reduced TOW echelon
form.
(5)
Suppose V and W are two linear spaces, and there xa an surjective map from Vto W. then
the dimension of W is less than the dimension of V.
Problem 2. Find the invertible 3 matrix P. and 5x5 invertible matrix Q such that PMQ is
equal to
PMQ
where
M
Problem 3. Suppose Vist & dimensional linear space over R with chosen basis e1 e e
e4
)
Let
W {ae bez cea a-2b-d 2a +6-c-d = =
U = {a, ben des a d d a-b-c
ce3
(1) Find basa for W and U. (write in terms of €1,e2,ey,es)
(2) Find baza for W U (write in terms of
(3) Find a basis for wnu (wire in terms of €1 €2 €3 es)
(4) For each of the following w. Determine whether it is in subspace W or not. if is in W.
find the coordinate of these vectors with respect to your basis.
Problem 4. Suppose Vis 3-dimensional linear space with basis (e e2 es ) and T:V V
is linear transformation such that
Ter
And W = {ae + bez+cegla+b-c=0
(1) Find a basis for W
(2) Show that W is an invariant subspace of T.
(3) With respect to your basis on subspace, find the matrix representation of T/w. where Th
means restriction of Ton W.
(4) Find the kernel of T/W
(5) Find the image of T/w
(6) Show that W ker T2
Problem 5. Find an invertible matrix P. such that p-1MP xa an upper triangular matrix. Where
M
Problem 6. Suppose V M2x2(R) is all2x2 matrices over IR. Consider the linear transformation
T: V - V
A h
Suppose we have the bases:
E1
(1) Find the matrix representation of T with respect to the basis (E1 €2 €3 €4
(3) Find the eigenvalues, eigenvectors, the geometric and algebraic multiplicity of each eigenvalue
(4) Does this linear transformation diagonalizable? if it is find an eigenbasis. If not. explain
why.
Problem 7. Let (r41)(542)(243) € C} be a 5-dimensional
linear space over C. Substitute by 17 defines linear transformation T V - V
(1) Choose a basis for this linear space.
(2) Find the matrix representation form of this linear transformation.
(3) Find the eigenvalue, eigenvector, the geometric and algebraic multiplicity of each eigenvalue
(4) Does this linear transformation diagonalizable? if is find an eigenbasis If not. explain
why.
(5) Calculate 7101
Problem 8. Suppose A is an 2x2 matrix with trace equal to 1 and determinant equal to Shou
that = 1
Problem 9. Suppose A is an 3 x 3 matrix and tr(A)= 6. tr(A²) 14. and get(A) 6. Find the
the characteristic polynomial of A
Problem 10. Suppose V is e 20-dimensional linear space over c. and T V - V is a lineas
transformation With the characteriatic polynomial fr(x) (At 2)°( (1)(1-1)°(x-2)", and
suppose we Anow
dim(Im (T 21³ 16
dim/Im (T 1)= 19
dim(Im (T 19
dim(Im (T 21)-) 17
(1)
(2) dim(Im (T 27))
dim( Im (T -1)2)
dim(Im -21))
(3) tr(T2++T)
det(T2 -T)
(4)
For 21. find all its eigenvalue with its algebraic and geometric multiplicity, and
its minimal polynomial
13

These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction
of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice.
Unethical use is strictly forbidden.