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
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 le
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 +
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 th
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
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
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 s
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 argu
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
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 al
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
Let A =
0 2 1
2 0 4
3 1 0
1 1 2
3
4
1
0
M44 (R). To calculate det(A), we row-reduce A to upper-
triangular form, keeping track of the changes to the determinant as we go.
det(A) =
=
=
0 2 1
1 0 2
det
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 m
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 orthonorma
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
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 (an
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
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). Fo
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
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)
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)
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
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 p
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 position
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
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 placehold
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
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) i
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