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264
Instruction: Vectors in
2
We use the term vector to refer to a finite list of numbers. If the list includes just two
numbers, an ordered pair, we call the list a vector in 2 . In this course, we will deal exclusively
with vectors in 2 and commonly call them two-space vectors or just simply vectors.
Let z1 ∧ z2 be real numbers. Then, u , v , w , z , OA , and 0 below are all examples of
vectors and accepted notations for vectors.
⎛z ⎞
⎛ 0.2 ⎞
⎛ 2⎞
⎡3⎤
u = ⎢ ⎥ , v = −1, 4 , w = ⎜ ⎟ , z = ⎜ 1 ⎟ , OA = ⎜ ⎟ , and 0 = 0, 0
⎝ 0.3 ⎠
⎝3⎠
⎣ −2 ⎦
⎝ z2 ⎠
Two 2 vectors are equal if and only if their corresponding entries are equal. Hence, if
z = v from the examples above, then z1 = −1 and z2 = 4 . Given two 2 vectors like u and v ,
their sum is the vector u + v whose entries are the sums of the corresponding entries of u and
⎛3⎞
⎛ −1⎞
⎛ 3 + −1 ⎞ ⎛ 2 ⎞
v . Let u = ⎜ ⎟ and v = ⎜ ⎟ . Then, u + v = ⎜
⎟ = ⎜ ⎟ . Given a vector u and a real
⎝ −2 ⎠
⎝ 4⎠
⎝ −2 + 4 ⎠ ⎝ 2 ⎠
number c, then cu is a scalar multiple of u obtained by multiplying each entry in u by c. For
⎛3⎞
⎛ 3 ⎞ ⎛ 7 ⋅ 3 ⎞ ⎛ 21 ⎞
instance, if u = ⎜ ⎟ and c = 7 ,then cu = 7 ⎜ ⎟ = ⎜
⎟=⎜
⎟.
⎝ −2 ⎠
⎝ −2 ⎠ ⎝ 7 ⋅ ( −2 ) ⎠ ⎝ −14 ⎠
In summary then, we have the following definitions.
A two-space vector is an ordered pair of real numbers. If
both numbers are zero, we call the vector the zero vector.
Let u = u1 , u2 and v = v1 , v2 be two-space vectors and let k be any
real number. Then, ku = ku1 , ku2 , u + v = u1 + v1 , u2 + v2 , and
u − v = u1 − v1 , u2 − v2 . We call ku a scalar multiple of u . We call u + v
the sum (or the resultant) of the vectors u and v . We call u − v the
difference of the vectors u and v .
Consider a rectangular coordinate system in the plane. Since every point in the plane is
⎛a⎞
determined by an ordered pair, we can identify a geometric point ( a, b ) with a vector ⎜ ⎟ .
⎝b ⎠
Hence, we may regard 2 as the set of all vectors with two real number entries just as we regard
2
as the set of all points in the plane.
265
Instruction: Vectors in
n
In this lecture we admit that n × 1 matrices are vectors in
n
.
A vector is a n × 1 matrix with real number entries and is said
to be an element of the Cartesian product, n .
Previously, we visualized the product X × Y = 2 as the Cartesian plane (seen below)
and points in the plane represent elements in X × Y . For example, the ordered pair (2,2) is the
point shown on the Cartesian plane below.
Now, the ordered pair becomes a vector as defined above and is considered to be in 2 . We call
⎛ 2⎞
⎜ ⎟ a column vector and ( 2, 2 ) a row vector.
⎝ 2⎠
The definition expands to include Cartesian products of more than two sets. For instance,
the set of all vectors with three entries can represent the Cartesian product of sets X 1 , X 2 , X 3
such that X 1 = X 2 = X 3 = . The product X 1 × X 2 × X 3 is called "r-three" and written
refer to ordered triples as vectors and write them as 3 × 1 matrices as below.
3
. We
⎡ x1 ⎤
x = ⎢⎢ x2 ⎥⎥
⎢⎣ x3 ⎥⎦
If n is a positive integer, n denotes the set of all n real number entry vectors (or ordered
n-tuples) written as n × 1 matrices. Two vectors in n are equal if the corresponding entries are
equal. Any vector with all zero entries is called the zero vector and denoted 0 .
Addition of two vectors in n is defined entry by entry. Given two vectors u and v in
n
, we obtain their sum u + v by adding the corresponding entries of u and v as demonstrated
below.
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⎡u1 ⎤
⎡v1 ⎤
⎡u1 + v1 ⎤
⎢u ⎥
⎢v ⎥
⎢u + v ⎥
2⎥
2⎥
⎢
⎢
Let u =
and v =
, then u + v = ⎢ 2 2 ⎥ .
⎢ ⎥
⎢ ⎥
⎢
⎥
⎢ ⎥
⎢ ⎥
⎢
⎥
⎣u n ⎦
⎣vn ⎦
⎣un + vn ⎦
Given a vector u in n and a number c in , the scalar multiple of u by c is the vector
cu obtained by multiplying each entry in u by c as demonstrated below.*
⎡u1 ⎤
⎡c ⋅ u1 ⎤
⎢u ⎥
⎢c ⋅ u ⎥
Let u = ⎢ 2 ⎥ , then cu = ⎢ 2 ⎥ .
⎢ ⎥
⎢
⎥
⎢ ⎥
⎢
⎥
⎣u n ⎦
⎣ c ⋅ un ⎦
Given vectors u1 , u 2 ,… , u p in
c1u1 + c2u 2 +
n
and scalars c1 , c2 ,… , c p , the vector
+ cn u n is called a linear combination of u1 , u 2 ,… , u p with weights c1 , c2 ,… , c p .
To see an example of a linear combination, consider the vectors in
3
below.
⎡1 ⎤
⎡0 ⎤
⎡0 ⎤
⎢
⎥
⎢
⎥
Let e1 = ⎢0 ⎥ , e 2 = ⎢1 ⎥ , and e3 = ⎢⎢0 ⎥⎥ .
⎢⎣0 ⎥⎦
⎢⎣0 ⎥⎦
⎢⎣1 ⎥⎦
⎡3⎤
If b is a linear combination of vectors e1 , e 2 , and e3 with weights 3, –2, 5, then b = ⎢⎢ −2 ⎥⎥ as
⎢⎣ 5 ⎥⎦
below.
⎡1 ⎤
⎡0 ⎤
⎡0 ⎤ ⎡3 ⋅1 ⎤ ⎡ −2 ⋅ 0 ⎤ ⎡5 ⋅ 0 ⎤ ⎡3 ⎤ ⎡ 0 ⎤ ⎡0 ⎤ ⎡3 + 0 + 0 ⎤ ⎡ 3 ⎤
⎢
⎥
⎢
⎥
b = 3 ⎢0 ⎥ − 2 ⎢1 ⎥ + 5 ⎢⎢ 0 ⎥⎥ = ⎢⎢3 ⋅ 0 ⎥⎥ + ⎢⎢ −2 ⋅1 ⎥⎥ + ⎢⎢5 ⋅ 0 ⎥⎥ = ⎢⎢0 ⎥⎥ + ⎢⎢ −2 ⎥⎥ + ⎢⎢0 ⎥⎥ = ⎢⎢0 − 2 + 0 ⎥⎥ = ⎢⎢ −2 ⎥⎥
⎢⎣0 ⎥⎦
⎢⎣0 ⎥⎦
⎢⎣1 ⎥⎦ ⎢⎣3 ⋅ 0 ⎥⎦ ⎢⎣ −2 ⋅ 0 ⎥⎦ ⎢⎣5 ⋅1 ⎥⎦ ⎢⎣0 ⎥⎦ ⎢⎣ 0 ⎥⎦ ⎢⎣5 ⎥⎦ ⎢⎣0 + 0 + 5 ⎥⎦ ⎢⎣ 5 ⎥⎦
Instruction: Matrix Transformations
Now that we have the terminology of vectors at our disposal, we turn to a new concept.
We define the product of a m × n matrix and a vector in n .
*
The term scalar is synonymous with real number.
267
Let A be a m × n matrix with columns a1 , a 2 ,… , a n . Let x ∈ n , then the product
matrix Ax is the linear combination of the columns of A using the corresponding
entries in x as weights. In symbols,
Ax = [a1
a2
⎡ x1 ⎤
⎢x ⎥
a n ] ⋅ ⎢ 2 ⎥ = x1a1 + x2a 2 +
⎢ ⎥
⎢ ⎥
⎣ xn ⎦
+ xna n .
We will use the definition above to define matrix transformations.
Let A be a m × n matrix with real number entries. A matrix transformation is a
Ax that maps vectors in n to vectors in m .
function of the form f : x
To illustrate, consider Maria's Optical Lab, which produces two types of lenses, polycarbonate
and high-index plastic. The columns of matrix A represent the two products. The rows of matrix
A represent cost for materials, labor, and machine wear to produce a single lens of each type as
detailed below.
polycarbonate
↓ high-index plastic
↓
⎡ 0.75 0.40 ⎤
A = ⎢⎢ 2.25 2.00 ⎥⎥
⎢⎣ 0.05 0.05 ⎥⎦
← materials
← labor
← machine wear
If x is a production vector—that is, a vector whose first and second entries equals the number of
polycarbonate lenses and number of high-index plastic lens produced respectively, the function
f :x
Ax maps production vectors to total cost vectors—that is, vectors whose entries equal
the cost in materials, labor, and machine wear respectively. For instance, if Maria's Optical Lab
⎡100 ⎤
produces 100 poly-carbonate lenses and 200 high-index plastic lenses, then x = ⎢
⎥ , and the
⎣ 200 ⎦
function maps x to a total cost vector as below.
⎡100 ⎤
f :⎢
⎥
⎣ 200 ⎦
⎡0.75 0.40 ⎤
⎡0.75 ⎤
⎡0.40 ⎤ ⎡ 75 ⎤ ⎡ 80 ⎤ ⎡155 ⎤
⎢ 2.25 2.00 ⎥ ⋅ ⎡100 ⎤ = 100 ⎢ 2.25⎥ + 200 ⎢ 2.00 ⎥ = ⎢ 225⎥ + ⎢ 400 ⎥ = ⎢625 ⎥
⎢
⎥ ⎢ 200 ⎥
⎢
⎥
⎢
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥
⎦
⎢⎣0.05 0.05 ⎥⎦ ⎣
⎢⎣0.05 ⎥⎦
⎢⎣0.05 ⎥⎦ ⎢⎣ 5 ⎥⎦ ⎢⎣ 10 ⎥⎦ ⎢⎣ 15⎥⎦
Thus, the given production cost is $155.00 in materials, $625.00 in labor, and $15.00 in machine
wear.
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Note that in the example above, the function mapped a vector from 2 to a vector in 3 .
The ability of matrix transformations to map vectors from n to vectors in m should pique the
interest of anyone who has ever played a video game where the 2-dimensional screen displays
three-dimensional figures.
269
Instruction: Vectors and Matrix Transformations
Example 1
Matrix Transformations
1⎤
⎡
−5 − ⎥
⎛3⎞
⎢
Suppose f : x 6 Bx where B =
2 . Find the image of x = ⎜ ⎟ .
⎢
⎥
⎝ 4⎠
3⎦
⎣ 6
1⎤
⎡
⎛ 1⎞
−
3
5
−
−
−
5
⎛
⎞
⎛
⎞
⎢
2 ⎥ ⎜ ⎟ = 3⎜ ⎟ + 4 ⎜ 2 ⎟
⎢
⎥ ⎝ 4⎠
⎝ 6⎠ ⎜ 3 ⎟
⎣ 6 3⎦
⎝
⎠
⎛ −15 ⎞ ⎛ −2 ⎞
=⎜
⎟+⎜ ⎟
⎝ 18 ⎠ ⎝ 12 ⎠
1⎤
⎡
⎢ −5 − 2 ⎥ ⎛ 3 ⎞ = ⎛ −17 ⎞
⎢
⎥ ⎜⎝ 4 ⎟⎠ ⎜⎝ 30 ⎟⎠
6
3
⎣
⎦
Example 2
Matrix Transformations
Suppose T is a mapping from \ n to \ m . What are the dimensions of the corresponding
matrix?
Recall that the matrix must have as many columns as the input vector has entries. Thus, the
matrix must have n columns. The number of rows determines the number of entries in the
output vector, so the matrix must have m rows. The matrix is a m × n matrix.
270
Example 3
Matrix Transformations.
⎛1 ⎞
⎡3 − 2 0 ⎤
⎜ ⎟
Suppose M : x 6 Ax where A = ⎢
. Find the image of x = ⎜ 2 ⎟ .
⎥
⎣1 − 1 4 ⎦
⎜ ⎟
⎝ −6 ⎠
⎛1 ⎞
⎡3 − 2 0 ⎤ ⎜ ⎟ ⎛ 3 ⎞ ⎛ −2 ⎞
⎛0⎞
⎢1 − 1 4 ⎥ ⎜ 2 ⎟ = 1⎜1 ⎟ + 2 ⎜ −1 ⎟ + −6 ⎜ 4 ⎟
⎣
⎦⎜− ⎟ ⎝ ⎠ ⎝ ⎠
⎝ ⎠
⎝ 6⎠
⎛ 3 ⎞ ⎛ −4 ⎞ ⎛ 0 ⎞
= ⎜ ⎟+⎜ ⎟+⎜
⎟
⎝1 ⎠ ⎝ −2 ⎠ ⎝ −24 ⎠
⎛ −1 ⎞
=⎜
⎟
⎝ −25 ⎠
271
Problems
#1
Consider the function H : x 6 A⋅ x , If A is a m × 2 matrix, then what is the domain of H?
#2
⎡1 − 5⎤
⎡ −1⎤
Use the function f : x 6 A⋅ x where A = ⎢⎢ 4
3 ⎥⎥ to map x = ⎢ ⎥ .
⎣ 4⎦
⎢⎣ 2 − 1⎥⎦
G
G
⎡a
Consider the function g : x 6 A⋅ x where A = ⎢
⎣a
G ⎡1⎤
G ⎡ 4⎤
x = ⎢ ⎥ and A ⋅ x = ⎢ ⎥ ?
⎣1⎦
⎣ 2⎦
#3
b⎤
. What is the value of a and b if
− b ⎥⎦
#4
⎡5
Consider the function F : x 6 ⎢
⎣4
−1 ⎤
⎛3⎞
x
.
Find
the
image
of
⎜ ⎟.
− 2 ⎥⎦
⎝7⎠
#5
⎡4
Consider the function T : x 6 ⎢
⎣0
−2
−3
#1 \
2
⎡ −21⎤
#2 ⎢⎢ 8 ⎥⎥
⎢⎣ − 6 ⎥⎦
⎛ 12 ⎞
⎜ ⎟ ⎛ −6⎞
#5 T : ⎜ 3 ⎟ 6 ⎜
⎟
⎜ −1⎟ ⎝ −10 ⎠
⎝ ⎠
⎛ 12 ⎞
2⎤
⎜ ⎟
x . Find the image of ⎜ 3 ⎟ .
⎥
1⎦
⎜ −1 ⎟
⎝ ⎠
#3 a = 3 and b = 1
⎛3 ⎞ ⎛8 ⎞
#4 F : ⎜ ⎟ 6 ⎜ ⎟
⎝ 7 ⎠ ⎝ −2 ⎠
272
Suggested Homework from Blitzer
Blitzer buries his discussion of transformations using matrices inside section 6.3, but the content
does not deal with the topic in a direct manner.
Application Exercise
⎡1 0 ⎤
Let R : x 6 Ax where A = ⎢
⎥ . Then, R is a transformation that reflects any
⎣0 − 1⎦
vector in \ 2 over the x-axis. Suppose V : x 6 Bx is a transformation that reflects
any vector in \ 2 over the y-axis. Find B .