variable y = z + 2v. Iterate: on the next row
up, substitute expressions found 26 Chapter
One. Linear Systems in lower rows x + (z + 2v)
+ 2z + (v) + v = 0 and solve for the leading
variable x = z 2v. To finish, write the
solution in vector notation x y z

bn an Vectors are equal if they have the
same representation. We arent too careful
about distinguishing between a point and the
vector whose canonical representation ends at
that point. R n = cfw_ v1 . . . vn | v1, . . . ,
vn R And, we do addition and sca

the homogeneous system has a non-~0
solution, and thus by the prior paragraph has
infinitely many solutions). QED This table
summarizes the factors affecting the size of a
general solution. number of solutions of the
homogeneous system particular solution

left side gives (u1 v1) 2 + (u2 v2) 2 + (u3
v3) 2 = (u 2 1 2u1v1 + v 2 1 ) + (u 2 2 2u2v2
+ v 2 2 ) + (u 2 3 2u3v3 + v 2 3 ) while the
right side gives this. (u 2 1 + u 2 2 + u 2 3 ) + (v
2 1 + v 2 2 + v 2 3 ) 2 |~u | |~v | cos
Canceling squares u 2 1 ,

miles at 89 degrees east of north, and 6.5
miles at 31 degrees east of north. Find the
distance between starting and ending
positions. (Ignore the earths curvature.) 2.14
Find k so that these two vectors are
perpendicular. k 1 4 3 2.15 Describe the set
of

unknowns and solve it. (b) Solutions to the
system are choices that the farmer can make.
Write down two reasonable solutions. (c)
Suppose that in the fall when the crops
mature, the farmer can bring in revenue of
$100 per acre for corn, $300 per acre for

classes we can think of the matrices in a
class as derived by row operations from the
unique reduced echelon form matrix in that
class. Put in more operational terms,
uniqueness of reduced echelon form lets us
answer questions about the classes by
transla

interaction between the left and right. For
some intuition about that interaction, consider
this system with one of the coefficients left
unspecified, as the variable c. x + 2y + 3z = 1 x
+ y + z = 1 cx + 3y + 4z = 0 If c = 2 then this
system has no solut

unknowns system having (a) a one-parameter
solution set; (b) a two-parameter solution set;
(c) a three-parameter solution set. ? 2.32
[Shepelev] This puzzle is from a Russian website http:/www.arbuz.uz/ and there are many
solutions to it, but mine uses li

in the plane they generate. (Remark. They
could generate a degenerate plane a line or
a point but the statement remains true.)
2.36 Prove that, where ~u,~v R n are
nonzero vectors, the vector ~u |~u | + ~v |~v |
bisects the angle between them. Illustrate

the prior section we defined vectors and
vector operations with an algebraic
motivation; r v1 v2 ! = rv1 rv2 ! v1 v2 ! + w1
w2 ! = v1 + w1 v2 + w2 ! we can now
understand those operations geometrically.
For instance, if ~v represents a displacement
then 3

third truck driver pay for a sandwich, a cup of
coffee, and a doughnut? X 2.21 The Linear
Combination Lemma says which equations can
be gotten from Gaussian reduction of a given
linear system. (1) Produce an equation not
implied by this system. 3x + 4y =

b c = 2 2a + c = 3 a b = 0 (e) x + 2y z = 3 2x
+ y + w = 4 x y + z + w = 1 (f) x + z + w = 4 2x +
y w = 2 3x + y + z = 7 X 2.19 Solve each
system using matrix notation. Give each
solution set in vector notation. (a) 2x + y z =
1 4x y = 3 (b) x z = 1 y + 2

row of the second is a linear combination of
the rows of the first. Proof For any two
interreducible matrices A and B there is some
minimum number of row operations that will
take one to the other. We proceed by
induction on that number. In the base step,

are straight and planes are flat. We can easily
check from the definition that linear surfaces
have the property that for any two points in
that surface, the line segment between them
is contained in that surface. But if the linear
surface were not flat t

step so by the principle of mathematical
induction the proposition is true. QED This
shows, as discussed between the lemma and
its proof, that we can parametrize solution sets
using the free variables. We say that the set of
vectors cfw_c1~ 1 + + ck~ k |

are equal. How can things that are in different
places be equal? Think of a vector as
representing a displacement (the word vector
is Latin for carrier or traveler). These two
squares undergo displacements that are equal
despite that they start in differe

have the zero vector as their only solution. 3x
+ 2y + z = 0 6x + 4y = 0 y + z = 0 21+2 3x
+ 2y + z = 0 2z = 0 y + z = 0 23 3x + 2y
+ z = 0 y + z = 0 2z = 0 3.5 Example Some
homogeneous systems have many solutions.
One is the Section I. Solving Linear Sys

has after a Gaussian reduction. The solution
description has two parts, the particular
solution ~p and the unrestricted linear
combination of the ~ s. We shall prove the
theorem with two corresponding lemmas. We
will focus first on the unrestricted
combin

equations conflict. But the associated
homogeneous system does have a solution, as
do all homogeneous systems. x + z + w = 0 2x
y + w = 0 x + y + 3z + 2w = 0 21+2
1+3 2+3 x + z + w = 0 y 2z w = 0
0 = 0 In fact, the solution set is infinite. cfw_
1 2 1

1 1 8 = 2 1 3 1 0 5
3 4 4.5 = 2 4 0.5
1 0 5 As with the line, note that we
describe some points in this plane with
negative ts or negative ss or both. Calculus
books often describe a plane by using a single
linear equation. P = cfw_ x y z | 2x + y + z

other rows by moving ci~i to the left and
dividing by ci. Therefore we will have proved
the theorem if we show that in () all of the
coefficients are 0. For that we use induction on
the row number i. The base case is the first
row i = 1 (if there is no su

base case is that the matrix has n = 1 column.
If this is the zero matrix then its echelon form
is the zero matrix. If instead it has any nonzero
entries then when the matrix is brought to
reduced echelon form it must have at least
one nonzero entry, whic

solves the given system since for any equation
index i, ai,1(p1 + h1) + + ai,n(pn + hn) =
(ai,1p1 + + ai,npn) + (ai,1h1 + + ai,nhn)
= di + 0 = di where as earlier pj and hj are the jth components of ~p and ~h. QED The two
lemmas together establish Theorem

adding a multiple of one row to another ri +j
then only row j of B differs from the matching
row of G, and ~ j = ri + j, which is indeed a
linear combinations of the rows of G. Because
we have proved both a base step and an
inductive step, the proposition

the picture, including the parallelogram in the
plane that shows the sum of the vectors
ending at (1.5, 0, 1) and (1.5, 1, 0). The
endpoint of the sum, on the diagonal, is not (3,
1, 1); what is it? 1.9 Show that the line
segments (a1, a2)(b1, b2) and (c1

equations? X 3.24 Prove that if ~s and ~t
satisfy a homogeneous system then so do
these vectors. (a) ~s +~t (b) 3~s (c) k~s + m~t
for k, m R Whats wrong with this argument:
These three show that if a homogeneous
system has one solution then it has many
so

(3, 3) to (2, 5) in R 2 (c) the vector from (1, 0, 6)
to (5, 0, 3) in R 3 (d) the vector from (6, 8, 8) to
(6, 8, 8) in R 3 X 1.2 Decide if the two vectors
are equal. (a) the vector from (5, 3) to (6, 2)
and the vector from (1, 2) to (1, 1) (b) the
vector

it another way and get y and w free. We close
with a recap. In Gausss Method we start with
a matrix and then derive a sequence of other
matrices. We defined two matrices to be
related if we can derive one from the other.
That relation is an equivalence re

~v |~u | |~v | ) (if either is the zero vector, we
take the angle to be right). 2.8 Corollary
Vectors from R n are orthogonal, that is,
perpendicular, if and only if their dot product
is zero. They are parallel if and only if their dot
product equals the

like the lines and planes of our past
experience. Rather, we must ensure that the
names suit the sets. While we cant prove that
the sets satisfy our intuition we cant prove
anything about intuition in this subsection
well observe that a result familiar fr

below). 2.5 Theorem (Triangle Inequality) For
any ~u,~v R n, |~u +~v | 6 |~u | + |~v | with
equality if and only if one of the vectors is a
nonnegative scalar multiple of the other one.
This is the source of the familiar saying, The
shortest distance betw