then 
.
For 1-9, decide if they are true or false. If true, give a proof. If false, give a counterexample to prove that it is false.
Q Any system of linear equations which has at least two solutions has an infinite number of solutions.
Q Any non-zero matrix is row-equivalent to a matrix with a ``1'' in the upper-left hand corner.
Q If A is an invertible matrix and then .
Q If a system of linear equations 
has a non-trivial solution,
then the matrix A is invertible.
Q If the main diagonal of an n x n matrix consists entirely of zeros, then the determinant is 0.
Q If 
and 
are linearly dependent
sets, then the set
is linearly dependent.
Q The vector space 
of all m x n matrices has dimension
mn.
Q There is no vector space of dimension 5 which has a subspace of dimension 7.
Q An n x n matrix A is invertible if and only if the column vectors
of 
(the transpose of A) are linearly independent
Q Let 
and
Compute via whatever method seems easiest: det
.
Q Find all solutions to:
for
,
,
and
.
Find a basis for the nullspace of 
.
Q Which of the following are vector spaces? For those which are not, state at least one axiom which fails to hold.
a) V is the set of polynomials which consist only of odd-powered terms, with the operation of addition being the usual addition of polynomials and the operation of scalar multiplication also being the usual multiplication.
b) V is the set of n by n matrices satisfying 
, with the
operations of addition and scalar multiplication being defined
as matrix addition and matrix scalar multiplication.
c) V is the set of real-valued functions f on the real line which
satisfy f(3)=f(-6), with addition defined as 
and scalar multipliction defined as 
Q Let be the matrix 
.
Compute the eigenvalues and eigenvectors of .
Draw a sketch of the flow assosiated to the linear
transformation given by left multiplication by .
Q Let B be the 4 x 7 matrix:
.
B is row equivalent to the 4 x 7 matrix
.
Find the rank and nullity of B. The row space of B is a ???
dimensional subspace of
??? dimensional Euclidean space.
The column space of R is a ??? dimensional subspace of
??? dimensional Euclidean space.
The column space of B is a ??? dimensional subspace of
??? dimensional Euclidean space.
The nullspace of B is a ??? dimensional subspace of
??? dimensional Euclidean space.
Give a basis for the column space for B.
Give a basis for the nullspace of B.
Give a basis for the row space of B. Give a basis for the column
space of R. What size matix is 
? What are the first row
and column
of 
? Are the rows of B a linearly independent set?
Q Give examples of two 3 x 5 matrices which have the same row space but different column spaces or prove that this never happens.
Q Show that there is a basis for 4 dimensional Euclidean space which contains the vector (1,1,1,1).
Q Let A be an n x n matrix. Which of the statements below are equivalent to A being invertible? Which are true for all invertible matrices A? Which imply that A is invertible?
A is row equivalent to the identity.
A has determinant 
.
The row vectors of A span n-dimensional Euclidean space.
The column vectors of A are linearly dependent.
The nullity of the transpose of A is the same as the nullity of A.
There is a linear transformation from 
to which corresponds
to the matrix A.
The system of equations 
has a solution.
A is an orthogonal matrix
Q
Give an example of a 3 x 3 matrix which has eigenvalues 
and
and which has eigenvectors (1,0,0) and (0,0,1) for the eigenvalue
or prove that there is no such matrix.
Q
Prove: For A and B n x n matrices, and P any n x n invertible matrix,
if 
, then A and B have the same eigenvalues.
(Hint: show that A and B have the same characteristic polynomial.)
Q Sketch the amplitude of the oscillation of an underdamped
mass spring system as a function of frequency, if the
system has a natural frequency 
and it is
driven by a sinusoidal force of frequency 
.
Q Solve the differential equation 
using power
series methods.
Q For what x does the following power series converge:
Q For what x does the following power series converge:
Q For what x does the following power series converge:
Q Find the Taylor series for 
the easy way.
Q There are 3 states a TV show can be in: very popular, somewhat popular, or unpopular. The chance that a very popular TV show stays very popular is 25%; the chance it becomes somewhat popular is 50% and the chance that it becomes unpopular is 25%. A somewhat popular show has the possibility of 10% of becoming very popular, 50% of staying somewhat popular, and 40% of becoming unpopular. And an unpopular TV show has a 5% chance of becoming very popular, a 10% chance of becoming somewhat popular, and an 85% chance of staying unpopular. Write down a Markov process that describes this system and compute the long-term fraction of shows that we expect to be very popular, somewhat popular and unpopular. The following commands have been executed by Mathematica and may be useful:
In[5]:= m = {{1/4,1/2,1/4},{1/10,1/2,2/5},{1/20,1/10,17/20}};
Eigensystem[m]
3 - Sqrt[3] 3 + Sqrt[3]
Out[6]= {{1, -----------, -----------}, {{1, 1, 1},
10 10
2 (-7 - 2 Sqrt[3]) -7 - 2 Sqrt[3]
{-11 - 2 Sqrt[3] + ------------------, -(--------------), 1},
Sqrt[3] Sqrt[3]
2 (-7 + 2 Sqrt[3]) -7 + 2 Sqrt[3]
{-11 + 2 Sqrt[3] - ------------------, --------------, 1}}}
Sqrt[3] Sqrt[3]
In[7]:= Eigensystem[Transpose[m]]
3 - Sqrt[3] 3 + Sqrt[3] 7 4
Out[7]= {{1, -----------, -----------}, {{--, --, 1},
10 10 65 13
-11 - 2 Sqrt[3] 8 (5 - 5 (11 + 2 Sqrt[3])) 5 - 5 (11 + 2 Sqrt[3])
{--------------- + --------------------------, -(----------------------), 1},
5 5 (-30 + 5 Sqrt[3]) -30 + 5 Sqrt[3]
-11 + 2 Sqrt[3] 8 (5 - 5 (11 - 2 Sqrt[3])) 5 - 5 (11 - 2 Sqrt[3])
{--------------- + --------------------------, -(----------------------), 1}}}
5 5 (-30 - 5 Sqrt[3]) -30 - 5 Sqrt[3]
Q Let
,
, and
.
Compute the matrix products AB, BA, AC, CA, BC, CB and the rank of each of
those matrices.