Solutions to Assignment 6
1. Find the coordinates of v with respect to the basis of the vector space
v. That is nd [v] .
(a) V = P2 (R), v = 1 x + 2x2 , = cfw_1 x, 1 + x + x2 , x + x2 .
1 2
3 4
(b) V = M22 (R), v =
1 1
0 0
=
1 0
1 0
,
0 0
1 1
,
1 0
0 1
So
Math 2121B Assignment 3
due Wednesday, February 13, 2013
Polynomials
Denitions:
1. The degree of a non-zero polynomial f (x) = n ai xi is the largest
i=0
n 0 such that an = 0. Then deg(f ) = n. The leading coecient of
f (x) is an . A polynomial is monic i
Math 2121 Assignment 5
1. In each case, show that V = U W and determine the projection map
pU,W : V V .
(a)
V = P3 (R), U = cfw_a + (a + b)x + bx2 |a, b R
W = cfw_cx + dx2 + (c + d)x3 |c, d R, v = 1 + 2x + 3x2 + 4x3
(b) V = M22 (R),
U=
a a
b b
: a, b R ,
Math 2121 Assignment 4
1. For the following matrices, nd the characteristic and minimal polynomials. Justify your answers.
(a)
A=
cos2 ()
cos() sin()
cos() sin()
sin2 ()
(b)
4 14 5
A = 1 4 2
1 6 4
2. For the following linear transformations, nd the minim
Review of Math 2120A and
Diagonalisation
Note: This doesnt include the material on Gaussian elimination or
on Matrix Operations which were in separate les.
1. Vector Spaces and Subspaces
Throughout this section, let F be a eld.
Denition. An (F)-vector spa
Math 2121 Assignment 2
due Friday, February 1, 2013
1. Suppose A Mnn (R) is a matrix satisfying A2 3A 4In = 0 which
is similar to an upper triangular matrix.
(a) What are the possible eigenvalues of A?
(b) Is A invertible? Why or why not? If so, nd a form
Math 2121 Assignment 1
due Friday, January 18, 2013
1. Let T : V W be a linear map, a basis for V and a basis for W .
Show that (R(T ) = R([T ] ). Deduce that ( )R(T ) maps R(T )
isomorphically onto R([T ] ).
2. For the following matrices, nd the eigenval
Solving Linear Systems of Equations
Denition 1 A system of m linear equations in n variables with coecients
in a eld F is
n
aij xj = bi , 1 i m
j=1
where aij F, bi F for all 1 i m, 1 j n. We want to solve
this system for all possible solutions x1 , . . .
1 Definition. A eld is a set F together with two associative, commutative binary operations,
one called addition and denoted by +, the other called multiplication, and denoted by juxtaposition
of arguments, such that the following properties hold:
(i)
(ii
Diagonalization examples
Let A =
2 1 1
1 2 1
1 1 2
M33 (R). Find a diagonal matrix D M33 (R) and an invertible matrix
P M33 (R) such that D = P AP 1 .
The solution to this problem requires that we nd an ordered basis for M31 (R) for which the
elements of
Matrix Operations
Denition 1 A eld is a set F closed under 2 operations : addition a, b
F implies a + b F and multiplication a, b F implies ab F . These
operations satisfy the following axioms for all a, b, c F
1. (commutativity) a + b = b + a
2. (associ
Assignment 10, due Tuesday April 8, at my oce
1. Let T : R4 R4 be the linear operator for which T (a, b, c, d) = (a + b, b c, a + c, a + d).
Find an ordered basis for the T -cyclic subspace of V with generator e1 .
Solution: T (e1 ) = (1, 0, 1, 1), T 2 (e
Assignment 10, due at the beginning of class, Monday, March 31
1. Let F be a eld. Prove that for any positive integer n, the determinant of any nn matrix
A whose entries are polynomials of degree at most 1 in the variable with coecients from
F is a polyno
Math 2121 Assignment 8
1. Orthogonally diagonalize
5 2 4
8 2
A = 2
4 2
5
2. Determine whether T : V V is normal, self-adjoint, skew-adjoint or
orthogonal/unitary. Find T and if possible an orthonormal basis of
eigenvectors of V for T . If T is orthogonal
Math 2121 Assignment 6
1. For the linear operator T , nd a Jordan canonical form J of T and a
Jordan canonical basis for T . Also nd a rational canonical form R
of T and a rational basis for T where T : M22 (R) M22 (R) is given
by
3 1
T (A) =
A AT
0 3
2.
Let F be a eld, and let V be an F -vector space. We wish to prove that if V has a linearly
independent set of any nite size n, then every spanning set of V has size at least n. We shall prove
this (and more) by induction on the size of a nite linearly ind
Solutions to Assignment 7
1. For the following matrices, nd the eigenvalues, a basis for each eigenspace
and the algebraic and geometric multiplicities of each eigenvalue. Determine whether or not the matrix is diagonalisable, and if so, nd an
invertible
Solutions to Midterm
1. (a) Is W = cfw_A M44 (R) : A2 = 0 a subspace of M44 (R)?
(b) Is U = cfw_f F(R, R) : f (x + ) = f (x) a subspace of F(R, R)?
Solution:
(a) No, W is not a subspace of M44 (R). For example, E12 , E21
2
M44 (R) but E12 + E21 W as (E12
Solutions to Assigment 5
1. (a) Is T : Mnn (R) R, T (A) = det(A), n 2 a linear transformation? If so, prove it. If not, give an explicit numerical counterexample.
(b) Repeat part (a) for T : F(R, R) R given by T (f ) = f (1).
(c) Suppose T : P2 (R) P2 (R)
Solutions to Assignment 3
1. Prove that V = R2 with addition dened as
(x1 , y1 ) (x2 , y2 ) (x1 + x2 , y1 + y2 + 1), (x1 , y1 ), (x2 , y2 ) V
and scalar multiplication dened as
a (x, y) (ax, ay + a 1), (x, y) V, a R
is a real vector space under these oper
Solutions to Assignment 4
1. (a) Let u, v, w be vectors in a vector space V . Show that Spancfw_u, v, w =
Spancfw_u v, u + w, w.
(b) If S, S are non-empty subsets of a vector space V such that
Span(S) = Span(S ) = V, must there be a vector common to
both
Solutions to Practice Problems for Final
1. Are the following maps linear transformations? Justify your answers.
(a) T : Mmn (R) R, T (A) = rank(A).
(b) S : F(R, R) R, S(f (x) = f (1).
Solution:
(a) T is not a linear transformation since 2n = T (In ) + T
Solutions to Practice Problems for Midterm I
Warning: These problems are only intended for practice for the midterm.
Do not assume the midterm will be modelled from these problems. You
should do the problems and use the solutions as a check on your answer
Solutions to Assignment 2
1. Determine all possible reduced row echelon forms of 2 by 4 matrices.
For each, clearly indicate the position of the leading 1 and put s in the
(possibly) non-zero positions. How many such forms are there? Note:
two forms are c
Solutions to Assignment 1
1. Let A be a 2 2 matrix. Show that AT A = AAT if and only if A is
a b
symmetric or A =
for some a and b.
b a
Solution: Let A =
a b
. The condition AT A = AAT implies that
c d
a2 + c2 ab + cd
ab + cd b2 + d2
=
a2 + b2 ac + bd
ac
Review of Summation Notation and its
Properties
n
For m, n Z, m n and ak , m k n, we write
k=m ak as
shorthand for the sum am + . . . + an . The variable k is a dummy
variable, it is just a placeholder.
Note that n cak = c( n ak ) if c is a constant not
Let F be a eld.
1 Definition. For any integer n 2, and any A Mnn (F ), for any i and j with 1 i n,
1 j n, let Ai,j denote the (n 1) (n 1) matrix obtained from A by deleting row i and column
j.
We dene the determinant of a square matrix with entries from F
Row reduced echelon form
Throughout this note, let F denote an arbitrary eld.
1 Definition. For positive integers m and n, A Mmn (F ) is said to be row reduced echelon
(we shall write rre for short) if either A = O or there is an integer k with 1 k n and
Math 2121 Assignment 7
1. (a) Find an orthonormal basis for the subspace W = Spancfw_ of V =
C[/2, /2] where is the linearly independent set cfw_sin(x), cos(x), x.
/2
and the inner product is given by f, g = /2 f (t)g(t)dt. Find
the closest vector to 1 in