Excess Burden Practice Problems (Part 1)
1. A friend says that a lump sum tax generates no excess burden because it does not lead to changes in
behavior. Is this statement true or false? Explain.
a. What is the definition of excess burden?
b. Using a grap
0 1 2 3 6 1 3 8 7 1 0 = 2 3 6
1 3 8 7 1 0 and from the right. 2 3 6
1 3 8 7 1 0 1 0 0 0 1 0 0 0 1 =
2 3 6 1 3 8 7 1 0 In short, an identity
matrix is the identity element of the set of nn
matrices with respect to the operation of
matrix multiplication.
same rank. There is only one rank zero matrix.
The other two classes have infinitely many
members; weve shown only the canonical
representative. One nice thing about the
representative in Theorem 2.6 is that we can
completely understand the linear map whe
because the matrix is nonsingular Corollary
IV.3.23 says there are elementary reduction
matrices such that Rr R1 M = I with r > 1.
Elementary matrices are invertible and their
inverses are also elementary so multiplying
both sides of that equation from th
equivalent if one can be converted to the
other by a sequence of row reduction steps,
while two matrices are matrix equivalent if
one can be converted to the other by a
sequence of row reduction steps followed by a
sequence of column reduction steps.
Cons
0 1 0 0 1 0 0 1 0
= 0 y 0 which matches our
intuitive expectation. The picture above
showing the figure walking out on the line
until ~vs tip is overhead is one way to think of
the orthogonal projection of a vector into a
line. We finish this subsection
model to ensure that the solution has a
desired accuracy. 4.7 Lemma A matrix H is
invertible if and only if it can be written as the
product of elementary reduction matrices. We
can compute the inverse by applying to the
identity matrix the same row steps
Matrix Multiplication We can consider matrix
multiplication as a mechanical process, putting
aside for the moment any implications about
the underlying maps. The striking thing about
this operation is the way that rows and
columns combine. The i, j entry
bases for the spaces. B = h 1 1 1 , 0
1 1 , 0 0 1 i C = h1 + x, 1 x, x2 i D =
h 1 0 0 0 , 0 2 0 0 , 0 0 3 0 , 0 0 0 4 i (a) Give the
formula for the composition map g h: R 3
M22 derived directly from the above
definition. (b) Represent h and g with respe
(3) If H ki+j G then Ci,j(k)H = G. Proof
Clear. QED 3.21 Example This is the first
system, from the first chapter, on which we
performed Gausss Method. 3x3 = 9 x1 + 5x2
2x3 = 2 (1/3)x1 + 2x2 = 3 We can reduce it with
matrix multiplication. Swap the first
changes a representation with respect to the
basis h~ 1, . . . , ~ i, . . . , ~ j, . . . , ~ ni into
one with respect to this basis 254 Chapter
Three. Maps Between Spaces h~ 1, . . . , ~ j, .
. . , ~ i, . . . , ~ ni. ~v = c1 ~ 1 + + ci ~ i
+ + cj~ j + + c
constructive it not only says the bases
change, it shows how they change. 1.21 Let V,
W be vector spaces, and let B, B be bases for
V and D, D be bases for W. Where h: V W is
linear, find a formula relating RepB,D(h) to
RepB, D (h). X 1.22 Show that the c
not orthogonal. B = h 4 2 , 1 3 i ~ 1 ~ 2 We
will derive from B a new basis for the space
h~1,~2i consisting of mutually orthogonal
vectors. The first member of the new basis is
just ~ 1. ~1 = 4 2 ! 270 Chapter Three. Maps
Between Spaces For the second me
is the 22 identity matrix, with 1s in its 1, 1
and 2, 2 entries and zeroes elsewhere; see
Exercise 34). (b) Let p(x) be a polynomial p(x) =
cnx n + + c1x + c0. If T is a square matrix we
define p(T) to be the matrix cnT n + + c1T +
c0I (where I is the app
symmetric if each i, j entry equals the j, i entry
(that is, if the matrix equals its transpose).
Show that the matrices HHT and HTH are
symmetric. (c) Show that the inverse of the
transpose is the transpose of the inverse. (d)
Show that the inverse of a
Gauss-Jordan reduction. We have already seen
how to produce a matrix that rescales rows,
and a row swapper. 3.16 Example Multiplying
by this matrix rescales the second row by
three. 1 0 0 0 3 0 0 0 1 0 2 1 1 0
1/3 1 1 1 0 2 0 = 0 2 1 1 0 1 3 3 1 0
2 0 3.1
sentence holds because matrix-vector
multiplication represents a map application
and so RepB,D(id) RepB(~v) = RepD(id(~v) ) =
RepD(~v) for each ~v. For the second sentence,
with respect to B, D the matrix M represents a
linear map whose action is to map e
Sample Practice Problems for Introductory Chapters
Introduction to Public Finance: Efficiency and Equity
1. The first fundamental theorem states that, under some conditions, the equilibrium in a
competitive market would be pareto efficient. Briefly explai
Tax Incidence Practice Problems
1. In 2013, the city of Cologne, Germany, instituted a pleasure tax. Among other things, the tax
applied to massage parlors, table-dancing clubs, and brothels. Many sex workers complained
that the tax was unjust because it
invertible they are square, and because their
product is defined they must both be nn. Fix
spaces and bases say, R n with the standard
bases to get maps g, h: R n R n that are
associated with the matrices, G = RepEn,En (g)
and H = RepEn,En (h). Consider h
gives some nice properties and more are in
Exercise 25 and Exercise 26. 2.12 Theorem If F,
G, and H are matrices, and the matrix products
are defined, then the product is associative
(FG)H = F(GH) and distributes over matrix
addition F(G + H) = FG + FH an
of the result. 1 2 3 4 5 6 7 8 9 0 1
0 0 0 0 = 0 1 0 4 0 7 Section IV.
Matrix Operations 235 3.4 Example Rescaling
unit matrices simply rescales the result. This is
the action from the left of the matrix that is
twice the one in the prior example. 0 2 0
0
proj[~2] (~ 3) . . . ~k = ~ k proj[~1] (~
k) proj[~k1] (~ k) form an
orthogonal basis for the same subspace. 2.8
Remark This is restricted to R n only because
we have not given a definition of orthogonality
for other spaces. Proof We will use induction
to
equals ~vi+1? If so, what is the earliest such i?
Section VI. Projection 269 VI.2 Gram-Schmidt
Orthogonalization The prior subsection
suggests that projecting ~v into the line
spanned by ~s decomposes that vector into
two parts proj[~s] (~p) ~v ~v proj[~s
think of orthogonal projection into a line is to
have the person stand on the vector, not the
line. This person holds a rope looped over the
line. As they pull, the loop slides on the line.
When it is tight, the rope is orthogonal to the
line. That is, we
sided inverse if and only if it is both one-toone and onto. The appendix also shows that if
a function f has a two-sided inverse then it is
unique, so we call it the inverse and write f
1 . In addition, recall that we have shown in
Theorem II.2.20 that if
calculate H = RepB, D (h) either by directly
using B and D , or else by first changing bases
with RepB,B (id) then multiplying by H =
RepB,D(h) and then changing bases with
RepD,D (id). H = RepD,D (id) H RepB,B
(id) () 2.1 Example The matrix T = cos(/6)
matrix just given m n p q! 1 1 2 1 ! = 1 0 0 1!
by using Gausss Method to solve the resulting
linear system. m + 2n = 1 m n = 0 p + 2q = 0 p
q = 1 Answer: m = 1/3, n = 1/3, p = 2/3, and q
= 1/3. (This matrix is actually the two-sided
inverse of H; the ch
exercises.) Here is another property of matrix
multiplication that might be puzzling at first
sight. (a) Prove that the composition of the
projections x, y : R 3 R 3 onto the x and y
axes is the zero map despite that neither one
is itself the zero map. (b