INTRODUCTORY LINEAR
ALGEBRA
Nguyen Minh Trang
✓ Textbook: Larson Ewards, Falvo; Elementary
Linear Algebra, 6th edition
✓ 3 credits (45 periods)
Chapter 1: Systems of Linear Equations,
Matrices and Determinants
Chapter 2: Vector Spaces and Euclidean
Spaces
Chapter 3: Linear Transformations
Chapter 4: Eigenvalue and Eigenvectors
What is Linear Algebra?
Problem 1: Suppose you travel on an airplane between two cities that are
5000 kms apart. If the trip one way against a headwind takes 25/4 hours
and the return trip the same day in the direction of the wind takes only 5
hours, can you find the ground speed of the plane and the speed of the
wind, assuming that both remain constant?
PROBLEM 2: THERE ARE SOME TOY HORSES AND TOY CLOWNS ON
SANTA’S SLEIGH. TOGETHER THEY HAVE 45 HEADS AND 148 FEET. FIND
THE NUMBER OF HORSES AND THE NUMBER OF CLOWNS?
Chapter 1: Systems of Linear Equations,
Matrices and Determinants
1.1. Introduction to Systems of Linear
Equations
1.2. Matrices
1.3. Determinants
1.1. Introduction to Systems of Linear Equations
Linear Equations in n variables
a1 x1 + a2 x2 +
+ an xn = b
• x1, x2, …, xn : variables
• a1, a2, …, an : coefficients
• b : constant
Example:
1) x1 + 2 x2 − 10 x3 + x4 = 10
1
2) x + y − z = 2
2
3) xy + z = 2
4) e x − 2 y = 4
Systems of Linear Equations
a11 x1 + a12 x2 + + a1n xn = b1
a x + a x + + a x = b
21 1 22 2
2n n
2
am1 x1 + am 2 x2 + + amn xn = bm
A system of LEs is called consistent if it has at least
one solution and inconsistent if it has no solution
Example:
x + y = 3
1)
x − y = −1
x + y = 3
2)
2 x + 2 y = 6
x + y = 3
3)
x + y = 1
Solving a system of Linear Equations
Which system is easier to solve algebraically?
x − 2 y + 3z = 9
1) − x + 3 y = −4
2 x − 5 y + 5 z = 17
x − 2 y + 3z = 9
2) y + 3 z = 5
z=2
(2) is in row-echelon form. To find its solution, we use
a procedure called back-substitution.
Operations that lead to equivalent systems of
equations
➢Interchange two equations
➢Multiply an equation by a nonzero constant
➢Add a multiple of an equation to another equation
Example: Solve the system
x − 2 y + 3z = 9
1) − x + 3 y = −4
2 x − 5 y + 5 z = 17
𝑥1 − 3𝑥2 + 𝑥3
2) ቐ2𝑥1 − 𝑥2 − 2𝑥3
𝑥1 + 2𝑥2 − 3𝑥3
=
=
=
1
2
−1
* Gaussian Elimination and Gauss-Jordan Elimination
Definition of a Matrix: If m and n are positive integers,
then an 𝑚 x 𝑛 matrix is a rectangular array
a11
a
21
a31
am1
a12
a22
a32
a13
a23
a33
am 2
am 3
a1n
a2 n
a3n
amn
in which each entry aij of the matrix is a number.
Examples: Determine the size of the matrix
1 2
1) 3 −4
0 1
−4
6
2
2 −1
2)
1 0
5) 2 1
4
2
−7 4
1 3
3) 4 1
2 1
1
2
4)
−1
3
If m = n, the matrix is called square of order n and
the entries a11, a22, a33, …, ann are called the main
diagonal entries
Examples
1
1)
3
2
4
1
2) 0
0
−1 0
1 2
0 1
3
1
3)
0
0
4
−1
1
0
5
0
2
1
6
−5
2
1
a11 x1 + a12 x2 + + a1n xn = b1
a x + a x + + a x = b
21 1 22 2
2n n
2
am1 x1 + am 2 x2 + + amn xn = bm
Coefficients and Augmented matrix of the system
a11
a
A = 21
...
am1
a12
a22
...
am 2
... a1n
... a2 n
... ...
... amn
a11
a
A = 21
am1
a12
a22
a1n
a2 n
am 2
amn
b1
b2
bm
Example:
Coefficient matrix
1 −4 3
−1 3 −1
2 0 4
x − 4 y + 3z = 5
− x + 3 y − z = −3
2 x
− 4z = 6
Augmented matrix
1 −4 3 5
−1 3 −1 −3
2 0 −4 6
Gaussian Elimination
1. Write the augmented matrix.
2. Use elementary row operations to rewrite the augmented
matrix in row-echelon form.
3. Write the system of linear equations corresponding to the
matrix in row-echelon form, and use back-substitution to
find the solution.
* Elementary row operations
-
Interchange two rows
Multiply a row by a nonzero constant
Add a multiple of a row to another row
* Row-Echelon Form
1
𝑎) 0
0
2 −1 4
1 0
3
0 1 −2
1 −5 2
0 0 1
𝑏)
0 0 0
0 0 0
−1 3
3 −2
1
4
0
1
1
0
c)
0
0
0
1
0
0
0 −1
0 2
1 3
0 0
Examples: Using elementary row operations to solve the
system
− 3 z = −2
x
b) 3 x + y − 2 z = 5
2 x + 2 y + z = 4
𝑥1 − 𝑥2 + 2𝑥3 = 4
𝑥1 +
𝑥3 = 6
c)
2𝑥1 − 3𝑥2 + 5𝑥3 = 4
3𝑥1 + 2𝑥2 − 𝑥3 = 1
➢Gauss – Jordan elimination
Gaussian elimination >> row-echelon form
Gauss-Jordan elimination >> reduced row-echelon form
Example:
− 3z = −2
x
3x + y − 2 z = 5
2 x + 2 y + z = 4
➢ Homogeneous Systems of Linear Equations
𝑎11 𝑥1 + 𝑎12 𝑥2 + ⋯ + 𝑎1𝑛 𝑥𝑛 = 0
𝑎21 𝑥1 + 𝑎22 𝑥2 + ⋯ + 𝑎2𝑛 𝑥𝑛 = 0
……
𝑎𝑚1 𝑥1 + 𝑎𝑚2 𝑥2 + ⋯ + 𝑎𝑚𝑛 𝑥𝑛 = 0
The system always has the solution zero, called
trivial (or obvious) solution.
𝑥
−
𝑥
+
3𝑥
=
0
1
2
3
Example: Solve the system ቊ
2𝑥1 + 𝑥2 + 3𝑥3 = 0
Theorem: Every homogeneous system of linear
equations is consistent. Moreover, if the system has
fewer equations than variables, then it must have an
infinite number of solutions.
Applications of Systems of Linear Equations
* Polynomial Curve Fitting
Problem: Suppose a collection of data is represented by n
points (𝑥𝑖 , 𝑦𝑖 ), i=1,2,…,n, in the 𝑥𝑦-plane. We have to find
a polynomial function of degree n-1
P( x) = a0 + a1 x + a2 x + ... + an−1 x
2
n −1
whose graph passes through all given points. This procedure
is called polynomial curve fitting.
Example: Determine the polynomial
𝑝 𝑥 = 𝑎0 + 𝑎1 𝑥 + 𝑎2 𝑥 2
whose graph passes through the points (2, 5), (3, 2), (4, 5).
* Network Analysis
• Networks composed of branches and junctions are used as
models in many diverse fields such as economics, traffic
analysis, and electrical engineering.
• In such models, the total flow into a junction is equal to the
total flow out of the junction
Examples: 1) Set up a system of linear equations to
represent the network shown below
2) Determine the currents I1, I2, and I3 for the electrical network
shown below
Two properties of electrical networks known as Kirchhoff’s laws:
- All the current flowing into a junction must flow out of it
- The sum of the products IR around a closed path is equal to
the total voltage in the path.
1.2. Matrices
𝑎11
𝑎21
𝐴 = 𝑎𝑖𝑗 𝑚×𝑛 = …
𝑎𝑚1
𝑎12
𝑎22
…
𝑎𝑚2
With a square matrix 𝐴 = 𝑎𝑖𝑗
𝑛×𝑛
𝑎11
𝑎12
𝑎22
…
…
…
𝑎1𝑛
𝑎2𝑛
…
𝑎𝑛𝑛
𝑎11
𝑎21
…
𝑎𝑛1
upper triangular matrix
𝑎22
…
𝑎𝑛2
… 𝑎1𝑛
… 𝑎2𝑛
…
…
… 𝑎𝑚𝑛
…
…
𝑎𝑛𝑛
below triangular matrix
𝑎11
1
𝑎22
𝐼𝑛 =
…
𝑎𝑛𝑛
diagonal matrix
1
…
1
identity matrix
1.2.1. Operations with matrices
Let 𝐴 = 𝑎𝑖𝑗
•
𝑚×𝑛
; 𝐵 = 𝑏𝑖𝑗
𝑚×𝑛
.
Equality of matrices
𝐴 = 𝐵 if 𝑎𝑖𝑗 = 𝑏𝑖𝑗 for 1 ≤ 𝑖 ≤ 𝑚, 1 ≤ 𝑗 ≤ 𝑛
•
Matrix addition
𝐴 + 𝐵 = 𝑎𝑖𝑗 + 𝑏𝑖𝑗
•
𝑚×𝑛
Scalar Multiplication: with k is a scalar,
𝑘𝐴 = 𝑘𝑎𝑖𝑗
𝑚×𝑛
•
Transpose of the matrix 𝐴 = 𝑎𝑖𝑗
𝑚×𝑛
is 𝐴𝑇 = 𝑎𝑗𝑖
𝑛×𝑚
•
Matrix Multiplication: for 𝐴 = 𝑎𝑖𝑗
A𝑃 = 𝑐𝑖𝑗
𝑚×𝑛
and 𝑃 = 𝑝𝑖𝑗
𝑚×𝑝
n
where cij = aik pkj = ai1 p1 j + ai 2 p2 j + ... + ain pnj
k =1
Examples: Find 𝐴𝐵 𝑇 𝑎𝑛𝑑 𝐵 𝑇 𝐴𝑇 , where
2
𝐴 = −1
0
1 −2
3
0
3 and 𝐵 = 2
−2 1
3
1
−1
0
𝑛×𝑝
1.2.2. The Inverse of a Matrix
Definition: An 𝑛 × 𝑛 matrix A is invertible
(nonsingular) if there exists an 𝑛 × 𝑛 matrix B such
that
AB = BA = In
The inverse of A denoted by A-1 = B
Examples: 1) Show that B is the inverse of A, where
−1 2
A=
−
1
1
2) Find the inverse of the matrix
1 −2
B=
1
−
1
1 4
A=
−
1
3
Finding the Inverse of a Matrix by Gauss – Jordan
Elimination
1. Write the matrix [A|I ]. This process is called adjoining
matrix I to matrix A.
2. Use elementary row operations to rewrite [A|I ] in the
form [I|B], then A-1 = B.
Examples: Find the inverse of the matrix
1 −1 0
A = 1 0 −1
−6 2 3
3 1 1
A = 1 3 1
1 1 3
1.3. Determinants
1.3.1. The Determinant of a Matrix
o 𝐴 = 𝑎𝑖𝑗
2×2
𝑎11
det 𝐴 = 𝐴 = 𝑎
21
𝑎12
𝑎22 = 𝑎11 𝑎22 − 𝑎21 𝑎12
o 𝐴 = 𝑎𝑖𝑗
𝑛×𝑛
• The minor 𝑀𝑖𝑗 of 𝑎𝑖𝑗 is the determinant of the matrix
obtained by deleting the ith row and jth column of A.
• The cofactor 𝐶𝑖𝑗 is given by 𝐶𝑖𝑗 = (-1)i+j𝑀𝑖𝑗
n
det( A) = aij Cij = ai1Ci1 + ai 2Ci 2 +
+ ainCin
det( A) = aij Cij = a1 j C1 j + a2 j C2 j +
+ anj Cnj
j =1
n
i =1
Remark
• The row (column) containing the most zeros is usually the
best choice for expansion by cofactors.
• If A is a triangular matrix of order n, then its determinant
is the product of the entries on the main diagonal
det(A) = a11a22…ann
Example: 1) Find determinant of the matrix
1 −2
𝑥 𝑦 1
−1 1
𝐴= 2 3 1
A=
0 2
0 −1 1
3 4
2) Find 𝑥 such that
𝑥
0
0
0
1
2
0
0
0
−1
𝑥−1
0
3 0
0 2
0 3
0 −2
3
0
=0
4
𝑥+2
1.3.2. Properties of a Determinant
1) det(A) = 0 if
• an entire row (column) consists of zeros
• two rows (columns) are equal
• one row (column) is a multiple of another
Example:
0 0 0
2 4 −5 = 0
1 6 7
1 2 −3
2 −1 −6 = 0
−2 0 6
1 −2 4
0 1 −2 = 0
1 −2 4
2) Elementary Row Operations and Determinants
Let A be square matrix, if B is obtained from A by
• interchanging two rows (columns) of A, then
det(B) = - det(A)
• adding a multiple of a row (column) of A to another, then
det(B) = det(A)
• multiplying a row (column) of A by a nonzero constant c,
det() = cdet(A)
Example: Which of two determinants shown below is
easier to evaluate?
1 −2 3 1
4 −6 3 2
A=
;
−2 4 −9 −3
3 −6 9 2
1 −2 3 1
0 2 −9 −2
B =
0 0 −3 −1
0 0 0 −1
Elementary row (column) opreations
• Interchanging two rows of a matrix changed the sign of its
determinant.
• Adding a multiple of one row to another did not change the
determinant.
• Multiplying a row by a nonzero constant multiplied the
determinant by that same constant.
Example: Evaluate the following determinant
1 −2
3 −4
A=
3 6
4 5
7 9
5 5
1 −1
3 2
3) Determinants with Matrix Operations
det(AB) = det(A)det(B)
det(A-1) = 1/det(A)
det(cA) = cndet(A)
det(A) = det(AT)
1.3.3. Applications of Determinants
• A square matrix A is invertible (nonsingular) if and only if
det(A) ≠ 0, and
1
−1
T
A =
C
det( A)
where 𝐶 = 𝑐𝑖𝑗 ; 𝑐𝑖𝑗 = (−1)𝑖+𝑗 𝑀𝑖𝑗
Example: Using the Adjoint of a matrix to find A-1
−1 3 2
A = 0 −2 1
1 0 −2
• Cramer’s rule: If a system of n linear equations in n
variables has a nonzero determinant |A|, then the solution
of the system is given by
det( A1 )
det( A2 )
x1 =
, x2 =
,
det( A)
det( A)
det( An )
, xn =
det( A)
where the ith column of Ai is the column of constants in the
system of equations
Example: Use Cramer’s Rule to solve the system
− x + 2 y − 3z = 1
+z=0
2 x
3 x − 4 y + 4 z = 2
• Area of the triangle whose vertices are 𝑥1 , 𝑦1 , 𝑥2 , 𝑦2 ,
𝑥3 , 𝑦3
𝑥 𝑦 1
1 1
± 𝑥2
2 𝑥
3
1
𝑦2
𝑦3
1
1
Example: Find the area of the triangle whose vertices are
(1, 1), (2, 4), and (4, 2).
• Volume of the tetrahedron whose vertices are
𝑥1 , 𝑦1 , 𝑧1 , 𝑥2 , 𝑦2 , 𝑧2 , 𝑥3 , 𝑦3 , 𝑧3 , (𝑥4 , 𝑦4 , 𝑧4 )
x1
1 x2
V =
6 x3
x4
y1
y2
y3
y4
z1
z2
z3
z4
1
1
1
1
Example: Find the volume of the tetrahedron whose
vertices are (0,4,1), (4,0,0), (3,5,2), and (2,2,5)
Exercises
1) Evaluate
3 1
2 1 1
a)
.
2
1
3
0
1
1 0
3
1
d)
1
1
1
3
1
1
1
1
3
1
1
1
1
3
3 1 1 1 1 −1
b) 2 1 2 . 2 −1 1
1 2 3 1 0 1
0
1
e)
1
1
1
0
a
b
1
a
0
c
1
b
c
0
2 1 1
c) 3 1 0
0 1 2
2
2) Evaluate the determinant by expanding the forth
column
2
1
1
1
1
2
1
1
1
1
2
1
x
y
z
t
3) Find the inverse of the matrix (if there exists)
1 1 2
a) A = 2 3 2
1 3 −1
2 −1
1 1
b) A =
−1 2
0 1
0 3
2 −1
3 1
2 1
4) Solve the systems of linear equations
2 x1 − x2 − x3 = 4
a ) 3 x1 + 4 x2 − 2 x3 = 11
3 x − 2 x + 4 x = 11
2
3
1
x1 + 2 x2 + 3 x3 − 2 x4 = 6
2 x − x − 2 x − 3x = 8
1 2
3
4
b)
3 x1 + 2 x2 − x3 + 2 x4 = 4
2 x1 − 3 x2 + 2 x3 + x4 = −8
5) By Gauss - Jordan elemination or the adjoint of a matrix, find
the inverse of the matrix as shown below (show your work!)
1 2 3
A = 0 1 −1
2 2 2
6) By Gauss - Jordan elemination or the adjoint of a matrix, find
the inverse of the matrix as shown below (show your work!)
1 1 1
A = 1 2 3
−1 −1 −2
CHAPTER 2: Vector Spaces and Euclidean Spaces
2.1. Vector Spaces
2.2. Subspaces of Vector Spaces
2.3. Spanning Sets and Linear Independence
2.4. Basis and Dimension
2.5. Coordinates and Change of Basis
2.6. Euclidean Spaces
Orthonormal Bases: Gram-Schmit Process
2.1. Vector Spaces
A vector is
• characterized by two
quantities (length and
direction)
• represented by a directed
line segment.
Example: use a directed
line segment to represent
each vector in the plane
a) u = (2, 3)
b) v = (-1, 2)
A vector in the plane
𝑥 = (𝑥1 , 𝑥2 )
𝑥1 , 𝑥2 ∶ components of the vector
Let u= 𝑢1 , 𝑢2 and v=(𝑣1 , 𝑣2 ) be two vectors in the plane.
• Equality: u = v ⇔ 𝑢1 = 𝑣1 , 𝑢2 = 𝑣2
• Addition: u + v = (𝑢1 + 𝑣1 , 𝑢2 + 𝑣2 )
Example: Find the sum of
a) u = (1, 4), v = (2, -2)
b) u = ( 3, -2), v = ( -3, 2)
c) u = (2, 1), v = (0, 0)
• Scalar multiplication:
𝑐v = 𝑐𝑣1 , 𝑐𝑣2 ; where 𝑐 is a scalar
Properties of Vector Addition and Scalar Multiplication in the Plane
Definition of a Vector Space
Let V be a set on which two standard operations are defined. If
the listed axioms are satisfied then V is called a vector space.
For every u, v, and w in V and every scalar c and d;
1. u + v is in V
Closure under addition
2. u + v = v + u
Commutative property
3. u + (v + w) = (u + v) +w
Associative property
4. u + 0= u
Additive identity
5. u + (-u)= 0
Additive inverse
6. cu is in V
Closure under scalar multiplication
7. c(u + v) = cu + cv
Distributive property
8. (c + d)u = cu + du
Distributive property
9. c(du) = (cd)u
Associative property
10. 1(u) = u
Scalar identity
Examples of vector space
1) 𝑅 2 = {v=(𝑣1 , 𝑣2 ): 𝑣1 , 𝑣2 ∈ 𝑅}
with the standard operations.
2) 𝑅 𝑛 = {v=(𝑣1 , 𝑣2 , … , 𝑣𝑛 ): 𝑣1 , 𝑣2 , … , 𝑣𝑛 ∈ 𝑅}
with the standard operations.
3) The set of all mxn matrices with the standard operations.
4) The set Pn of all polynomials of degree n or less with the
standard operations.
2.2. Subspaces of vector spaces
Definition: A nonempty subset W of a vector space V is called
a subspace of V if W is a vector space under the operations of
addition and scalar multiplication defined in V.
Test for a subspace: If W is a nonempty subset of a vector
space V, then W is a subspace of V if and only if the following
closure conditions hold
1. u,v ∈ W ⇒ u + v ∈ W
2. u ∈ W ⇒ cu ∈ W, for any scalar c
Examples: The set 𝑊 = 𝑥1 , 𝑥2 : 𝑥1 ≥ 0 𝑎𝑛𝑑 𝑥2 ≥ 0 , with the
standard operations, is not a subspace of 𝑅2
Examples: Which of the subsets below is a subspace of 𝑅3
a) 𝑊 =
b) 𝑊 =
𝑥1 , 𝑥2 , 1 : 𝑥1 and 𝑥2 are real numbers
𝑥1 , 𝑥2 + 𝑥3 , 𝑥3 : 𝑥1 and 𝑥3 are real numbers
2.3. Spanning sets and Linear Independence
Linear Combination of Vectors
A vector v in a vector space V is called a linear combination of the
vectors u1, u2, …, uk in V if v can be written in the form
v = c1u1 + c2u2 +
+ ck uk
where c1, c2, …, ck are scalars
Examples: find a linear combination
a) Write the vector w = (1, 1, 1) as a linear combination of
vectors in the set S
S = {v1=(1, 2, 3), v2= (0, 1, 2), v3=(-1, 0, 1)}
b) If possible, write the vector w = (1, -2, 2) as a linear
combination of vectors in the set S above.
Spanning set of a Vector space
Let S = {v1, v2, …, vk} be a subset of a vector space V. The set S is
called a spanning set of V if every vector in V can be written as a
linear combination of vectors in S.
In such cases it is said that S spans V.
Examples:
a) S = {v1(1, 0, 0), v2(0, 1, 0), v3(0, 0, 1)} spans R3
b) S = {v1(1, 2, 3), v2(0, 1, 2), v3(-2, 0, 1)} spans R3
c) S = {v1(1, 2, 3), v2(0, 1, 2), v3(-1, 0, 1)} doesn’t span R3
Span of a Set
𝐒 = v1 , v2 , … , vk is a set of vectors in a vector space V,
span 𝐒 = span v1 , v2 , … , vk
= 𝑐1 v1 + 𝑐2 v2 + ⋯ + 𝑐𝑘 vk ∶ 𝑐1 , 𝑐2 , … , 𝑐𝑘 are real numbers
If span(S) =V it is said that V is spanned by S or S spans V
Span(S) is a Subspace of V
span 𝐒 is the smallest subspace of V that contains S, in the sense that
every other subspace of V that contains S must contain span(S).
Example:
a) S =
1,0,0 , 0,1,0 , 0,0,1
b) S = 1, 𝑥, 𝑥 2 spans P2
spans 𝑅3
Linear Independence and Linear Dependence
S={v1, v2, …, vk} is a set of vectors in a vector space V.
If the vector equation
c1v1 + c2v2 + … + ckvk = 0
• has only the trivial solution c1 = c2 = … = ck = 0, then S is called
linearly independent
• also has nontrivial solutions, then S is called linearly dependent.
Examples: Test for linear (in)dependence
1) S = {v1 = (1, 2, 3), v2 = (0, 1, 2), v3 = (-2, 0, 1)} 𝑅3
2) S = v1 = 1,1, −2 , v2 = 2,5, −1 , v3 = (1,0,1) 𝑅3
1 1 0 0
2 1 3 0 1 0
0 1 3 1
3) S =
,
,
M 22 4) S = , , , M 41
1
−1
−1 0
0 1 2 1 2 0
0 2 −2 2
2.4. Basis and Dimension
Basis
• A set of vectors S ={v1, v2, …, vk} in a vector space V is called a
basis for V if: (1) S spans V; (2) S is linearly independent.
Examples:
a) S= { v1=(1, 0), v2=(0, 1)} is a basis for R2
b) S = {v1=(1, 2, 3), v2=(0, 1, 2), v3=(-2, 0, 1)} is a basis for R3
• If a vector space V has one basis with n vectors, then every basis for
V has n vectors
Examples:
c) S={v1=(3, 2, 1), v2=(7, -1, 4)} isn’t a basis for R3
d) S={x + 2, x2, x3 - 1, 3x + 1, x2 - 2x + 3} isn’t a basis for P3
Dimension of a Vector Space
• If a vector space V has a basis consisting of n vectors, then the
number n is called the dimension of V, denoted by dim(V) = n.
If V consists of the zero vector alone, the dimension of V is defined
as zero, namely dim(V) = 0.
Examples:
a) The dimension of Rn with the standard operations is n.
b) The dimension of Pn with the standard operations is n + 1.
c) The dimension of Mm,n with the standard operations is mn.
Remark: If W is a subspace of an n-dimensional vector space V, then
the dimension of W is less than or equal to n.
Examples: Find a basis for and the dimension of a subspace
d) W = {(d, c - d, c): c and d are real numbers} R3
e) W = {(2b, b, 0): b is a real number} R3
2.5. Coordinates and Change of Basis
Coordinates with respect to a basis
B= {u1, u2, …, un} is an ordered basis for a vector space V
x ∈V
c1
c
x = c1u1 + c2u2 + … + cnun x B = 2
cn
E.g. 1: Find the coordinates of x = (1,2,-1) in R3 with respect to
a) the standard basis S = {u1=(1,0,0), u2=(0,1,0), u3=(0,0,1)}
b) the ordered basis B = {u1=(1,0,1), u2=(0,-1,2), u3=(2,3,5)}
E.g. 2: Find the coordinates of p = 3x3 - 2x2 + 4 in P3 with respect
to the standard basis S = {1, x, x2, x3}
Change of Basis
B = {u1, u2,…, un}, B’ = {v1, v2,…, vn}: ordered bases for a
vector space V. x ∈ V.
[x]B = [c1 c2 … cn]T;
[x]B’ = [d1 d2 … dn]T
• The transition matrix from B’ to B is the matrix P such that
[x]B = P[x]B’
• P is invertible and P-1 is the transition matrix from B to B’.
• To find P:
[vj]B
j = 1,2,…,n
PB '→ B = v1 B v2 B
vn B
Example: Let B={u1(1,0,0), u2(0,1,0), u3(0,0,1)} and
B’={v1(1,0,1), v2(0,-1,2),v3(2,3,-5)} be two given bases in R3
a) Find the transition matrix from B’ to B
b) Find the transition matrix from B to B’
c) Find [x]B when [x]B’ = [1 2 -1]T
2.5. Euclidean Space
Rn with vector addition and scalar multiplication is
called Euclidean n-space
• Dot product of u = (u1, u2, …, un) and v = (v1, v2, …, vn)
u.v = u , v = u1v1 + u2v2 +
+ u n vn
E.g: Find the dot product of u=(1,2,0,-3) and v=(3,-2,4,2) in R4
Note: If u.v = 0, then u and v are orthogonal in Rn
E.g: a) Prove that u=(3,2,-1,4) and v=(1,-1,1,0) are orthogonal in R4
b) Determine all vectors in R2 that are orthogonal to u=(4,2)
• Length (or magnitude, norm) of a vector v = (v1, v2, …, vn)
v =
v, v = v12 + v22 +
+ vn2
If v = 1, v is called a unit vector
• For v ≠ 0, u = v/ v has the length 1 and the same direction as v.
This vector u is called the unit vector in the direction of v
Note: - The dot product in Rn is called the Euclidean inner product.
- A vector space V with an inner product is called an inner
product space.
Orthonormal Bases
• A set S of vectors in an inner product space V is called orthogonal if
every pair of vectors in S is orthogonal
• If, in addition, each vector in the set S is a unit vector, then S is
called orthonormal.
Exp: Determine whether S is orthogonal, orthonormal, or neither
a) S = {v1=(-4, 6), v2=(5, 0)} R2
b) S = {v1=(2,3,2,-2), v2 =(1,0,0,1), v3 =(-1,0,2,1),
v4 =(-1,2,-1,1)} R4
2 2 2 2
1 1
2 2 1
3
c) S = v1 =
,
, 0 , v2 = −
,
,
, v3 = , − , R
3
3 3 3
2 2
6 6
Theorem: If V is an inner product space of dimension n, then any
orthogonal set of n nonzero vectors is a basis for V
Exp: Prove that the set S is a basis for R4
S = {v1=(2,3,2,-2), v2=(1,0,0,1), v3=(-1,0,2,1), v4=(-1,2,-1,1)}
Gram – Schmidt orthonormalization Process
Let B = {v1, v2,…,vn} be a basis for an inner product space V. Then
B’ = {u1, u2,…, un} is an orthonormal basis for 𝑉, where
w1
v1
u1 =
=
w1
v1
w2
u2 =
, where w2 = v2 − v2 , u1 u1
w2
w3
u3 =
, where w3 = v3 − v3 , u1 u1 − v3 , u2 u2
w3
wn
un =
, where wn = vn − vn , u1 u1 − ... − v3 , un −1 un −1
wn
Example: Apply the Gram-Schmidt process to the basis below
1) B = {v1=(1, 1), v2=(0, 1)} R2
2) B = {v1=(1, 1, 0), v2=(1, 2, 0), v3=(0, 1, 2} R3
Exercise:
1) Given the set S = {v1(1, 3, 1, 3), v2(0, 1, 1, 0), v3(-3, 0, 6, -1),
v4(3, 4, -2, 1)} in R4
a) Determine whether the S is linearly independent or not?
b) Find a basis for and the dimension of a subspace of R4
spanned by the set S
2) Given two sets B ={u1(1, 3, 4), u2(2, -5, 2), u3(-4, 2, -6)} and
B’={v1(1, 2, -2), v2(4, 1, -4), v3(-2, 5, 8)} in R3
a) Prove that the B and B’ are bases in R3
b) Find the transition matrix from B to B’
c) Find the transition matrix from B’ to B
d) Find [x]B when provided with [x]B’=[1 -1 2]T
3) Find a basis for and the dimension of the solution space of the
system of linear equations
x1 + 2 x2 + 3x4 = 0
x3 + x4 = 0
4) Apply the Gram - Schmidt process to the basis in R3 shown
below
S = v1 (1,1,1), v2 (1, 2,1), v3 (1, 2, 2)
CHAPTER 3: LINEAR TRANSFORMATIONS
3.1. Introduction to Linear Transformations
3.2. The Kernel and Range of a Linear Transformation
3.3. Matrices for Linear Transformations
3.4. Transition Matrices and Similarity
KEY WORDS
• LINEAR TRANSFORMATIONS
• IMAGE AND PREIMAGE
• DOMAIN AND CODOMAIN
• KERNEL AND RANGE
• NULLITY AND RANK
• STANDARD MATRIX FOR A LINEAR TRANSFORMATION
• MATRIX OF A LINEAR TRANSFORMATION RELATIVE TO BASES B AND B’
3.1. Introduction to Linear Transformations
Functions map a vector space V into a vector space W denoted by
T: V → W
V: domain
W: codomain
If v is in V, w is in W such that
T(v) = w,
w: image of v under T
set of all w in W: range of T.
set of all v in V such that T(v) = w: preimage of w
Example: For any vector v = (x, y) in R2, let T: R2 → R2 be defined by
T(x, y) = (x - y, x + 2y)
a) Find the image of v = (-1, 2)
b) Find the preimage of w = (-1, 11)
Definition of a Linear Transformation
Let V and W be vector spaces. The function T: V → W is called a
Linear Transformation of V into W if the following two properties are
true
1. T(u + v) = T(u) + T(v) for u, v in V
2. T(cu) = cT(u) for u in V and any scalar c
Example: Show that the function is a linear transformation from R2 into
R2
T(x, y) = (x - y, x +2y)
Note: Most of the common functions studied in calculus are not linear
transformations
T(x) = sinx
T(x) = x2
Properties of Linear Transformations
T: V → W is a linear transformation; u, v ∈ V
1. T(0) = 0
2. T(-v) = -T(v)
3. T(u - v) = T(u) - T(v)
4. If v = c1v1 + c2v2 + … + cnvn, then
T(v) = c1T(v1) + c2T(v2) + … + cnT(vn)
E.g. 1: Let T: R3 → R3 be a linear transformation such that T(1, 0, 0) =
(2, -1, 4), T(0, 1, 0) = (1, 5, -2), T(0,0,1)=(0,3,1). Find T(2, 3, -2).
E.g. 2: The function T: R2 → R3 is defined as follows
3 0
v1
T (v) = Av = 2 1
v2
−1 −2
a) Find T(2, -1)
b) Show that T is a linear transformation from R2 to R3
3.2. The Kernel and Range of a Linear Transformation
Kernel of a Linear Transformation
•
Definition: Let T: V → W be a linear transformation. Then the set
of all vectors v in V that satisfy T(v) =0 is called the kernel of T
and is denoted by Ker(T)
Ker(T)= v V : T(v) = 0
Example: Find the kernel of a linear transformation T: R3→R2
defined by T(x) = Ax, where
1 −1 −2
A=
−
1
2
3
• The kernel of linear transformation T: V → W is a subspace of the
domain V. The dimension of ker(T) is called the nullity of T
dim(Ker(T) = nullity(T)
Example: Let T: R5 → R4 be defined by T(x) = Ax, where x is in R5 and
1
2
A=
−1
0
2 0
1 3
0 −2
0 0
1 −1
1 0
0 1
2 8
Find a basis for and the dimension of ker(T) as a subspace of R5
The Range of a Linear Transformation
• The range of T: V → W is the set of all vectors w in W that are
images of vectors in V denoted by range(T).
range(T)= T(v) : v V
• The range of T: V → W is a subspace of W. The dimension of
range(T) is called the rank of T denoted by rank(T)
rank(T) = dim(range(T))
• To find a basis for the range of a linear transformation defined by
T(x) = Ax, observe that the range consists of all vectors b such that
the system Ax = b is consistent.
By writing the system
a1n x1 b1
a11 a12
a
x b
a
a
22
2n 2
21
= 2
a
a
a
m2
mn xn
m1
bn
in the form
a11
a12
a1n b1
a
a
a b
Ax = x1 21 + x2 22 + + xn 2 n = 2 = b
am1
am 2
amn bn
So, b is in the range of T if and only if b is a linear combination of the
column vectors of A. So the column space of the matrix A is the same
as the range of T.
• If 𝑇: 𝑅 𝑛 → 𝑅 𝑚 be the linear
transformation given 𝑇 𝑥 = 𝐴𝑥.
Then the column space of A is
equal to the range of T and
𝐧𝐮𝐥𝐥𝐢𝐭𝐲 𝐓 + 𝐑𝐚𝐧𝐤 𝐓 = 𝐧
Examples: The linear transformation
T is represented by T(v)= Av. Find a
basis for the range(T)?
1 2 0
1) A =
0
1
−
1
1 2 −1 4
3 1 2 −1
2) A =
−4 −3 −1 −3
−1 −2 1 1
3.3. Matrices for linear transformations
The standard basis for Rn is represented by
B = e1 , e2 ,
1 0
0 1
, en =
,
,
0 0
0
0
,
1
Let 𝑇: 𝑅 𝑛 → 𝑅𝑚 be a linear transformation such that
𝑇 𝑒1
𝑎11
𝑎12
𝑎1𝑛
𝑎21
𝑎22
𝑎2𝑛
= ⋮ , 𝑇 𝑒2 = ⋮ , …., 𝑇 𝑒𝑛 = ⋮
𝑎𝑚1
𝑎𝑚2
𝑎𝑚𝑛
Then matrix A is called the standard matrix for T as shown below
T(v) = Av for every v ∈ 𝑅𝑛
𝑎11
𝑎21
𝐴= ⋮
𝑎𝑚1
𝑎12
𝑎22
⋮
𝑎𝑚2
⋯ 𝑎1𝑛
⋯ 𝑎2𝑛
⋮
⋮
⋯ 𝑎𝑚𝑛
Example:
a) Find the standard matrix for 𝑇: 𝑅3 → 𝑅2 defined by
𝑇 𝑥, 𝑦, 𝑧 = (𝑥 − 2𝑦, 𝑦 + 3𝑧)
b) Find the standard matrix for 𝑇: 𝑅3 → 𝑅3 defined by
𝑇 𝑥, 𝑦, 𝑧 = (5𝑥 + 3𝑦 − 𝑧, 4𝑦 − 2𝑧, 5𝑥 + 𝑦)
• Let V and W be finite-dimensional vector spaces with bases
B = { u1, u2, …, un} V and B’ = { v1, v2, …, vm} W
If T: V → W is a linear transformation such that
a11
a12
a1n
a
a
a
21
22
2n
T
(
u
)
=
,
T
(
u
)
=
,
,
T
(
u
)
=
1 B ' 2 B ' n B '
am1
am 2
amn
then the mxn matrix whose n columns correspond to [T(ui)]B’
a11
a
A = 21
am1
a12
a22
am 2
a1n
a2 n
amn
is such that [T(u)]B’ = A[u]B for every u in V.
A is called the matrix for T relative to the bases B and B’.
Remark
1) If B and B’ are the standard bases in V and W respectively, then A
is called the standard matrix for T
2) If V = W and B = B’ then A is called the matrix for T relative to B
and write A[u]B = [T(u)]B
E.g. 1: T: 𝑅2 → 𝑅3 via T(x, y) = (x + y, 2x – y, x) and the two bases
B={u1(1,2), u2(-1,1)} R2, B’={v1(1,0,1), v2(0,1,1), v3(1, 1, 0)} R2
a) Find the matrix for T relative to B and B’?
b) Using the matrix in (a) to find T(u) where u = (0, 3)?
E.g. 2: T: R3 → R2 via T(x, y, z) = (x - y, y - z) and the two bases
B={u1(1,1,1), u2(1,1,0),u3(1,0,0)} R3, B’={v1(1,2), v2(1,1)} R2
Find the matrix for T relative to B and B’?
3.4. Transition matrices and Similarity
The matrix for T: V → V depends on the basis for V. In other
words, the matrix for T relative to a basis B is different from the
matrix for T relative to another basis B’.
Definition of Similar matrices: For square matrices A and A’ of
order n, A’ is said to be similar to A if there exists an invertible
matrix P such that A’ = P-1AP
Theorem: T: Vn → Vn is a linear transformation. If A is the matrix
for T relative to B and A’ is the matrix for T relative to B’, then
A’ = P-1AP
Where P is the transition matrix from B’ to B
So, A’ is similar to A
Example 1: Suppose that
1 3 0
A = 3 1 0
0 0 −2
is the matrix for T: R3 → R3 relative to the standard basis. Find the
matrix for T relative to the basis
B’={v1(1, 1, 0), v2(1, -1, 0), v3(0, 0, 1)}
Exp2) Let B={u1(1, 0), u2(0, 1)} and B’={v1(-1, 2), v2(2, -2)} be
bases for R2, and let
−2 7
A=
−
3
7
be the matrix for T: R2→R2 relative to B. find A’?
Exp3) T: R2 → R3 is defined by T(x,y) = (x - y, 2x + 3y, -x)
a) Find a basis for ker(T) and nullity(T)
b) Find a basis for range(T) and rank(T)
c) Find the standard matrix for T
d) Find the matrix for T relative to B = {u1(1, -1), u2(0,1)} R2
and B’ ={v1(1, 1, 1), v2(1, 1, 0), v3(1, 0, 0)} R3
Exp 4) T: R3 → R3 is defined by
T(x,y,z) = (x - y + 2z, 2x + y - z, x + 2y + z)
and the basis B = {u1(1, 0, 1), u2(0, 2, 2), u3(1, 2, 0)}
Find the matrix for T relative to the basis B?
Exp 5) Let B = {u1(1, 0, 1), u2(0, 2, 2), u3(1, 2, 0)} and
B’ = {v1(1, 1, -1), v2(1, 0, 1), v3(0, 2, 0)} be bases for R3, and let
1 3 −1
A = 2 0 5
6 −2 4
be the matrix for T: R3 → R3 relative to B. find the matrix for T
relative to B’
Exercise1)
Given the two sets S = {u1(1, 1, 1), u2(1, 1,0), u3(1, 0, 0)}
and S’ = {v1(-6,-6, 0), v2(-2,-6, 4), v3(-2,-3, 7)} in R3
1) Find the transition matrix from S to S’
2) Find the transition matrix from S’ to S
3) Apply Gram - Schmidt process to the basis S for R3
Exercise2: find a basis for and the dimension of, the subspace of
R3 spanned by S
S = {u1(1,2,2), u2(-1,0,0), u3(1,1,1)}
Exercise3: find a basis for and the dimension of the solution
space of the following system
− x + y + z = 0
=0
3 x − y
2 x − 4 y − 5 z = 0
4 x − y + 2 z = 0
2 x + 3 y − z = 0
3x + y + z = 0
CHAPTER IV: Eigenvalues and Eigenvectors
4.1. Eigenvalues and Eigenvectors
4.2. Diagonalization
4.3. Applications
Keywords: eigenvalue, eigenvector, eigenspace, characteristic
equation, characteristic polynomial, multiplicity, diagonalizable,
diagonalization
4.1. Eigenvalues and Eigenvectors
Definitions
Let A be an nxn matrix. The scalar is called an eigenvalue of A if
there is a nonzero vector x such that Ax = x. The vector x is
called an eigenvector of A corresponding to
Example 1: for the matrix
2 0
A=
0
−
1
x1=(1, 0) is an eigenvector of A corresponding to the eigenvalue
1 = 2;
x2=(0,1); 2 = -1
Example 2: for the matrix
1 −2 1
A = 0 0 0
0 1 1
verify that x1=(-3, -1, 1) and x2 = (1, 0, 0) are eigenvectors of A
and find their corresponding eigenvalues
Finding Eigenvalues and Eigenvectors
Theorem: Let A be an nxn matrix
1) An eigenvalue of A is a scalar such that
det( A -I) = 0
2) The eigenvectors of A corresponding to are the nonzero
solutions of
(A-I )x = 0
In other words, the set of all eigenvectors of , together with the
zero vector, is a subspace of Rn. This subspace is called the
eigenspace of .
Example 1: find the eigenvalues and corresponding eigenvectors
of
2 −12
A=
1
−
5
2 1 0
A = 0 2 0
0 0 2
Note: If A is an nxn triangular matrix, then its eigenvalues are
the entries on its main diagonal.
2 0 0
A = −1 1 0
5 5 −3
−1
0
A=
0
0
0
2
0
0
0 0
0 0
0 0
0 −4
Example 2: find a basis for each of the corresponding
eigenspaces
3 −2 0
A = −2 3 0
0 0 5
3 1 1
A = 0 3 1
0 0 3
4.2. Diagonalization
Problem: For a square matrix A, does there exist an invertible
matrix P such that P-1AP is diagonal?
Remind: Two square matrices are similar if there exists an invertible
matrix P such that B = P-1AP
Definition of a diagonalizable matrix
An nxn matrix A is diagonalizable if A is similar to a diagonal
matrix, that is, there exists an invertible matrix P such that P-1AP
is a diagonal matrix
Example: verify that A is diagonalizable by computing P-1AP?
1 3
3 1
1) A =
, P=
−
1
5
1
1
2 3 1
1 1 5
2) A = 0 −1 2 , P = 0 −1 1
0 0 3
0 0 2
Problem
Which square matrices are
diagonalizable?
Theorem: An nxn matrix
A is diagonalizable if and
only if it has n linearly
independent eigenvectors
p1, p2, …, pn
Determine the matrix P?
P = [p1 p2 … pn]
Example
Show that the matrix A is diagonalizable
1 −1 −1
A = 1 3 1
−3 1 −1
Find a matrix P such that P-1AP is diagonal?
Theorem: If an nxn matrix A has n distinct eigenvalues, then the
corresponding eigenvectors are linearly independent and A is
diagonalizable.
Example: Determine whether the matrix A is diagonalizable
1 −2 1
A = 0 0 1
0 0 −3
Example: find an invertible matrix P such that P-1AP is diagonal
1 0
1) A =
6
−
1
2 0 −2
3) A = 0 3 0
0 0 3
1 0 0
2) A = 0 1 1
0 1 1
−1 4 0
4) A = 0 4 0
−3 1 3
Symmetric matrices and Orthogonal diagonalization
* A square matrix A is symmetric if it is equal to its transpose
𝐴 = 𝐴𝑇
Example: Determine whether the matrices are symmetric or not
0 1
1) 𝐴 = 1 3
−2 0
−2
0
5
3 2
2) 𝐴 = 1 −4
1 0
1
0
5
3) 𝐴 =
4 3
3 1
•
If A is an 𝑛 × 𝑛 symmetric matrix, then the following properties
are true
1. A is diagonalizable
2. All eigenvalues of A are real
3. If is an eigenvalue of A with multiplicity 𝑘, then has 𝑘
linearly independent eigenvectors. That is, the eigenspace of
has dimension 𝑘
Example: find the eigenvalues of the symmetric matrix
2 0 0
𝐴= 0 1 1
0 1 1
and determine the dimension of the corresponding eigenspaces
• A square matrix P is called orthogonal if it is invertible and if
𝑃−1 = 𝑃𝑡
Otherwise, the column vectors of P form an orthonormal set.
For example, the matrix 𝑃 =
0 1
is orthogonal
−1 0
Orthogonal diagonalization
Let A is an 𝑛 × 𝑛 symmetric matrix
1. Find all eigenvalues of A
2. Find a basis for each eigenspace, then apply the Gram-Schmidt
orthonormalization process
3. Set up the matrix P such that 𝑃−1 𝐴𝑃 = 𝑃𝑇 𝐴𝑃 = 𝐷
Example: find an orthogonal matrix P that orthogonally
diagonalizes
1) 𝐴 =
−2 2
2 1
2 0 0
2) 𝐴 = 0 1 1
0 1 1
2
2
3) 𝐴 = 2 −1
−2 4
−2
4
−1
4.3. Application: Quadratic form
Exercise 1: a linear transformation T: R3 → R3 is given by
T(x, y, z) = (x + 3y, y - 2z, -2x - 12z)
1) Find a basis for and the dimension of ker(T)?
2) Find the standard matrix for T?
3) Find a basis for and the dimension of range(T)?
4) Find the matrix A for T relative to B = {u1(1,1,1), u2(1,1,0),
u3(1,0,0)} R3
5) Use the matrix A to find T(1, -2, 0)?
Exercise 2: a linear transformation T: R2 → R3 is given by
T(x, y) = (x - y, 2x + 3y, -y)
1) Find the matrix A for T relative to B={u1(1,0), u2(1, 1)} R2
and B’={v1(1,0,1), v2(0,2,0), v3(-1,1,2)} R3
2) Use the matrix A to find T(1, -2)?
3) Assume that a matrix Q is the matrix for T relative to two bases
B R2 and B’ R3, where
1 −1
Q = 2 0 and B = {u1 (1, −1), u2 (1, 0)} R 2
−4 5
Find a basis B’ R3
0
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