Steven a leduc linear algebra (cliffs quick review) cliffs notes (1996)

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Cliffs Quick Revie w

Linear
Algebra
by
Steven A . Leduc

Series Edito r
Jerry Bobrow, Ph .D .

Wiley Publishing, Inc .


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Cliffs Quick Revie w

Linear
Algebra
by
Steven A . Leduc

Series Edito r
Jerry Bobrow, Ph .D .

Wiley Publishing, Inc .


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CliffsNotes TM Linear Algebra
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VECTOR A►LGEBRA

1

The Space R2
Vectors in R2
Position vectors

Vector addition
Vector subtraction
Scalar multiplication
Standard basis vectors in R2
The Space R 3
Standard basis vectors in R3
The cross product
The Space R"
The norm of a vector
Distance between two points
Unit vectors
The dot product
The triangle inequality
The Cauchy-Schwarz inequality
Orthogonal projections
Lines
Planes

1
3
7
8
11
12
13
16
17
19
22
23

25
26
27
31
32
36
39
43

's

47

MA
A►I~GEBItA
Matrices
Entries along the diagonal
Square matrices
Triangular matrices
The transpose of a matrix
Row an column matrices
Zero matrices
Operations with Matrices
Matrix addition

LINEAR ALGEBRA

47
49
50

50
51
52
53
53
53


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CONTENTS

Scalar multiplication
Matrix multiplication
Identity matrices
The inverse of a matrix

56
57
70
76

LINEAR SYSTEMS
Solutions to Linear Systems
Gaussian Elimination
Gauss-Jordan Elimination
Using Elementary Row Operations to Determine K 1

85
85

90
96
109

REAL EUCLIDEAN VECTOR SPACES
Subspaces of Rn
The Nullspace of a Matrix
Linear Combinations and the
Span of a Collection of Vectors
Linear Independence
The Rank of a Matrix
A Basis for a Vector Space
Orthonormal bases
Projection onto a Subspace
The Gram-Schmidt orthogonalization algorithm
The Row Space and Column Space of a Matrix
Criteria for membership in the column space
The Rank Plus Nullity Theorem
Other Real Euclidean Vector Space s
and the Concept of Isomorphism
Matrix spaces
Polynomial spaces
Function spaces

123
12 3
13 0

iv

134
137
145
150
157
160
16 7
17 6
17 7
18 3
18 9
18 9
19 2
194

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CONTENTS

THE DETE

ANT

Definitions of the Determinant
Method 1 for defining the determinant
Method 2 for defining the determinant
Laplace Expansions for the Determinant

Laplace expansions following row-reduction
Cramer's Rule
The Classical Adjoint of a Square Matrix

LINEAR TRANSFORMATIONS
Definition of a Linear Transformation
Linear Transformations and Basis Vectors
The Standard Matrix of a Linear Transformation
The Kernel and Range of a Linear Transformation
Injectivity and surj ectivity
Composition of Linear Transformations

197
19 7
19 7
20 6
220
23 4
23 6
24 1

251
25 1
260
263
27 2
28 0
28 5

EIGENVALUES AND EIGENVECTORS

293

Definition and Illustration o f
an Eigenvalue and an Eigenvector
Determining the Eigenvalues of a Matrix
Determining the Eigenvectors of a Matrix
Eigenspaces
Diagonalization

29 3
29 4
29 8
30 7
31 1

LINEAR ALGEBRA
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VECTOR ALGEBRA

It is assumed that at this point in your mathematical education, you are familiar with the basic arithmetic operations an d
algebraic properties of the real numbers, the set of which i s
denoted R . Since the set of reals has a familiar geometric depiction, the number line, R is also referred to as the real lin e

and alternatively denoted R ' ("R one") .

The Space R2
Algebraically, the familiar x -y plane is simply the collection of
all pairs (x, y) of real numbers. Each such pair specifies a
point in the plane as follows . First, construct two copies of the
real line—one horizontal and one vertical which intersect
perpendicularly at their origins ; these are called the axes .
Then, given a pair (xl , x2), the first coordinate, x l , specifies th e
point's horizontal displacement from the vertical axis, whil e
the second coordinate, x2, gives the vertical displacement fro m
the horizontal axis . See Figure 1 . Clearly, then, the order i n
which the coordinates are written is important since the point
(x l , x2) will not coincide generally with the point (x2, xl) .
To emphasize this fact, the plane is said to be the collection o f
ordered pairs of real numbers . Since it takes two real number s
to specify a point in the plane, the collection of ordered pair s
(or the plane) is called 2-space, denoted R 2 ("R two") .

LINEAR ALGEBRA
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VECTOR
ALGEBRA

;

(XI , X2 )

x2
xl

■ Figure 1

■

R2 is given an algebraic structure by defining two operation s
on its points . These operations are addition and scalar multiplication . The sum of two points x = (xl , xZ ) and x' = (x;, xZ )
is defined (quite naturally) by the equation
x+x' =(x l , x Z) + (x i, x2) = (xl +x i, xZ +x2 )
and a point x = (xl , x2 ) is multiplied by a scalar c (that is, by
a real number) by the rule
cx = c(xl , x2) = (cx1, cx2 )
Example 1 : Let x = (1, 3) and y = (—2, 5) . Determine th e
points x + y, 3x, and 2x — y .
The point x + y is (1, 3) + (—2, 5) = (—1, 8), and the poin t
3x equals 3(1, 3) = (3, 9) . Since —y = (—1)y = (2, -5) ,
2x—y=2x+(—y) = 2(1, 3) + (2, -5 )
= (2, 6) + (2, -5) = (4, 1 )

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VECTOR
ALGEBRA

By defining x — x' to be x + (—x'), the difference of tw o
points can be given directly by the equatio n
x - x' _

( .710

1 x2 ) - (x;,
,

JICZ)

_

(xi -

7Ci ,

x2

- JIC2 )

Thus, the point 2x — y could also have been calculated as follows :
2x—y = 2(1, 3)—(—2, 5) = (2, 6)—(—2, 5 )
= (2 — (—2), 6 — 5) = (4, 1)

111

Vectors in R 2. A geometric vector is a directed line segment
from an initial point (the tail) to a terminal or endpoint (th e

tip) . It is pictured as an arrow as in Figure 2 .
endpoint

("tip" )

initial point
("tail")
■ Figure 2

■

The vector from point a to point b is denoted ab . If a = (a l ,
a2) is the initial point and b = (b 1 , b2) is the terminal point ,
then the signed numbers b 1 — a1 and b2 — a2 are called th e
components of the vector ab . The first component, b 1 — a l ,
indicates the horizontal displacement from a to b, and the second component, b2 — a2, indicates the vertical displacement .
See Figure 3. The components are enclosed in parentheses t o
specify the vector ; thus, ab = ( b 1 — al, b2 — a2) .

LINEAR ALGEBRA

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VECTOR
ALGEBRA

■ Figure 3

■

Example 2 : If a = (4, 2) and b = (—5, 6), then the vector from
a to b has a horizontal component of -5 — 4 = -9 and a vertical component of 6 — 2 = 4 . Therefore, ab = (—9, 4), which i s
sketched in Figure 4 .
b=(—5,6)

ab = (—I, 4 )
4
-9

a=(4,2 )

■ Figure 4 ■

4

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VECTO R
ALGEBRA

Example 3 : Find the terminal point of the vector xy = (8, -7 )
if its initial point is x = (—3, 5) .
Since the first component of the vector is 8, adding 8 t o
the first coordinate of its initial point will give the first coordinate of its terminal point . Thus, y l = x i + 8 = -3 + 8 = 5 .
Similarly, since the second component of the vector is -7 ,
adding -7 to the second coordinate of its initial point will giv e

the second coordinate of its endpoint . This gives y 2 = xZ +
(—7) = 5 + (—'7) = -2 . The terminal point of the vector xy i s
therefore y = (y,, y2) = (5, -2) ; see Figure 5 .
x=(—3,5)

xy = (8, 7)

8

Y=
Y2 )
= (x 1 + 8, x 2 — 7)
=(—3+8, 5—7 )
= ( 5, — 2 )

■ Figure 5 ■
Two vectors in R2 are said to be equivalent (or equal) i f
they have the same first component and the same secon d
component . For instance, consider the points a = (—1, 1), b =
(l, 4), c = (l, 2), and d = (3, 1) . The horizontal component
of the vector ab is 1 — (—1) = 2, and the vertical component o f
ab is 4 — 1 = 3 ; thus, ab = (2, 3) . Since the vector cd has a

LINEAR ALGEBRA

5


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VECTO R

ALGEBRA

horizontal component of 3 – 1 = 2, and a vertical componen t
of 1 – (–2) = 3, cd = (2, 3) also . Therefore, ab = cd; see Figure 6 .
b=(1,4)

a=(—1, 1)

o

d = (3, 1 )

c= (1, 2)

■ Figure 6 ■
To translate a vector means to slide it so as to change its initia l
and terminal points but not its components . If the vector ab i n
Figure 6 were translated to begin at the point c = (1, 2), it
would coincide with the vector cd. This is another way to say
that ab = cd .
Example 4 : Is the vector from a = (0, 2) to b = (3, 5) equivalent to the vector from x = (2, -4) to y = (5, 1) ?
Since the vector ab equals (3, 3), but the vector xy equals
(3, 5), these vectors are not equivalent . Alternatively, if th e
vector ab were translated to begin at the point x, its termina l
point would then be (x l + 3, x 2 + 3) = (2 + 3, -4 + 3) = (5 ,
-1) . This is not the point y ; thus, ab ~ xy . ■

CLIFFS QUICK REVIEW

6

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VECTOR
ALGEBRA

Position vectors . If a vector has its initial point at the origin ,
the point 0 = (0, 0), it is called a position vector . If a position
vector has x = (x 1 , x2 ) as its endpoint, then the component s
of the vector Ox are x 1 — 0 = x 1 and x 2 — 0 = x 2 ; so Ox =
(x l , x2 ). If the origin is not explicitly written, then a positio n
vector can be named by simply specifying its endpoint ; thus, x
= (x 1 , x 2 ). Note that the position vector x with components x 1
and x2 is denoted (x 1 , x 2), just like the point x with coordinates
x 1 and x2 . The context will make it clear which meaning is in tended, but often the difference is irrelevant . Furthermore ,
since a position vector can be translated to begin at any othe r
point in the plane without altering the vector (since translatio n
leaves the components unchanged), even vectors that do not
begin at the origin are named by a single letter .
Example 5 : If the position vector x = (-4, 2) is translated s o
that its new initial point is a = (3, 1), find its new termina l
point, b.
If b = (b l , b2 ), then the components of the vector ab ar e
bl — 3 and b 2 — 1 . Since ab = x ,
(b l — 3, b 2 — 1) = (—4, 2) ~ (b i, b2) = (—1, 3 )
See Figure 7.

LINEAR ALGEBRA
7

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VECTOR
ALGEBRA

b = (-1, 3)

x= (-`1 , 2 )



■ Figure 7

a = (3,1 )

■

Vector addition . The operations defined earlier on points (xl ,
x2) in R 2 can be recast as operations on vectors in R 2 (calle d
2-vectors, because there are 2 components) . These operations
are called vector addition and scalar multiplication . The sum
of two vectors x and x' is defined by the same rule that gave
the sum of two points :
x + x ' = (x l , x 2 ) + (x1', x2) = (xl + x;, x 2 + x2 )
Figure 8 depicts the sum of two vectors . Geometrically, one o f
the vectors (x', say) is translated so that its tail coincides with
the tip of x . The vector from the tail of x to the tip of th e
translated x' is the vector sum x + x' . This process is often
referred to as adding vectors tip-to-tail.

8

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VECTOR
ALGEBRA

■ Figure 8 ■
Because the addition of real numbers is commutative, that is,
because the order in which numbers are added is irrelevant, i t
follows that
(x i +

x2

+ x)

= ( x 1 + xl , x2 +x2 )

This implies the addition of vectors is commutative also :
x+x'= x' + x
Thus, when adding x and x' geometrically, it doesn't matte r
whether x' is first translated to begin at the tip of x or x i s
translated to begin at the tip of x' ; the sum will be the same i n
either case .
Example 6 : The sum of the vectors x = (1, 3) and y = (-2, 5 )

is
x + y = (1 + (-2), 3 + 5) = (-1, 8 )
See Figure 9 .

LINEAR ALGEBRA

9


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VECTOR
ALGEBRA

x+ Y = (1 , 8)

Y=(2,5 )

■ Figure 9 ■
Example 7 : Consider the position vector a = (1, 3) . If b is th e
point (5, 4), find the vector ab and the vector sum a + ab .
Provide a sketch .
Since ab has horizontal component 5 — 1 = 4 and vertica l
component 4 — 3 = 1, the vector ab equals (4, 1) . So a + ab =
(1, 3) + (4, 1) = (5, 4), which is the position vector b . Figure
10 clearly shows that a + ab = b .

10

CLIFFS QUICK REVIEW

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VECTOR
ALGEBRA

ab = (4, 1)
i

b = (5, 4)

a = (1, 3)

■ Figure 10

■

Vector subtraction . The difference of two vectors is defined
in precisely the same way as the difference of two points . For
any two vectors x and x' in R2,
x — x' _ (xl , x2 ) — (x;, x2) _ (x1 — x;, x 2 — x2 )
With x and x' starting from the same point, x — x' is the vector that begins at the tip of x' and ends at the tip of x . Thi s
observation follows from the identity x' + (x — x') = x and the
method of adding vectors geometrically . See Figure 11 .

■ Figure 11

LINEAR ALGEBRA

■

11


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VECTOR
ALGEBRA

In general, it is easy to see tha t
ab = b — a
whether the letters on the right-hand side are interpreted a s
position vectors or as points . Figure 10 showed that a + ab =
b, which is equivalent to the statement ab = b — a, where a and
b are position vectors . Although this example dealt with a particular case, the identity ab = b — a holds in general .
Example 8 : The vector ab from a = (4, -1) to b = (—2, 1) i n
Figure 12 i s
ab=b—a=(—2, 1)—(4,—1)=(2—4, 1 +1)=(-6,2 )



,

■ Figure 12 ■
Scalar multiplication. A vector x is multiplied by a scalar c b y
the rule
cx = c(xI, x2 ) = (cxI, cx2 )

12

CLIFFS QUICK REVIEW

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VECTO R
ALGEBR A

If the scalar c is 0, then for any x, cx equals (0, 0)—the zero
vector, denoted 0. If c is positive, the vector cx points in th e
same direction as x, and it can be shown that its length is c
times the length of x . However, if the scalar c is negative, the n
cx points in the direction exactly opposite to that of the original x, and the length of cx is Id times the length of x . Some
examples are shown in Figure 13 :

■ Figure 13

■

Two vectors are said to be parallel if one is a positive scala r
multiple of the other and antiparallel if one is a negative scalar multiple of the other . (Note : Some authors declare two
vectors parallel if one is a scalar multiplepositive or negative—of the other.)
Standard basis vectors in R 2. By invoking the definitions o f
vector addition and scalar multiplication, any vector x = (x, ,
x2) in R2 can be written in terms of the standard basis vector s

LINEAR ALGEBRA

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VECTOR

ALGEBRA

(1, 0) and (0, 1) :
(x,, x2 )=(x„ 0)+(0, x2 )=x, (l, 0)+x2 (0, 1 )
The vector (1, 0) is denoted by i (or e,), and the vector (0, 1 )
is denoted by j (or e 2). Using this notation, any vector x in R2
can be written in either of the two form s
x = x l i +x2j or x = x l el + x2 e2
See Figure 14 .

j'

xli

x=x1i+x 2j
■ Figure 14 ■
Example 9 : If x = 2i + 4j and y = i — 3j, determine (and pro vide a sketch of) the vectors x and x + y .
Multiplying the vector x by the scalar

2

yields

x= (2i+4j) = ( . 2)i+(. .
The sum of the vectors x and y is

14

CLIFFS QUICK REVIEW

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VECTO R
ALGEBR A

x + y = (2i + 4j) + (i — 3j) _ (2 + 1)i + (4 — 3)j = 3i + j
These vectors are shown (together with x and y) in Figure 15 .
x=2i+4j
y=i—3j

2x=i+2j

x+y=3i+ j

Sr

= I - 3j

■ Figure 15

■

Example 10 : Find the scalar coefficients k, and k2 such that
k,(1, -3) + k2(—1, 2) = (—1, -2 )

The given equation can be rewritten as follows :
(lc, — k2 , -3k l + 2k2) = (—1, -2)
This implies that both of the following equations must be satisfied :
-3k1 + 2k2 = -2

(* )

Multiplying the first equation by 3 then adding the result t o
the second equation yields

LINEAR ALGEBRA

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VECTO R
ALGEBRA

Ski — 3k2

= -3

+2k2 =—2
—k2

= -5

Thus, k2 = 5 . Substituting this result back into either of th e
equations in (*) gives kl = 4 . ■

The Space R3
mutually perpendicular copies of the real line intersec t
at their origins, any point in the resulting space is specified b y
an ordered triple of real numbers (x 1 , x2 , x3). The set of al l

ordered triples of real numbers is called 3-space, denoted R 3
("R three") . See Figure 16 .
If three

■ Figure 16 ■
The operations of addition and scalar multiplication define d
on R2 carry over to R3:

16

CLIFFS QUICK REVIEW