Transcript File

Introduction
There are two types of physical quantities:
 Scalars = quantities that can be described by numerical
value alone (Ex: temperature, length, speed)
 Vectors = quantities that require both a numerical value
and direction (Ex: velocity, force,…)
Linear Algebra is concerned with 2 types of mathematical
objects, matrices and vectors. In this section we will
reciew the basic properties of vectors in 2D and 3D with
the goal of extending these properties in 𝑹𝒏
𝒗 = 𝑨𝑩
Continued on Next Slide 
Section 3.1 in Textbook
Definitions
 Vectors with the same length and direction are said to
be equivalent.
 The vector whose initial and terminal points coincide
has length zero so we call this the zero vector and
denote it as 0.
 The zero vector has no natural direction therefore we
can assign any direction that is convenient to us for the
problem at hand.
 The solutions to a system of linear equations in n
variables are n x 1 column matrices, the entries
representing values for each of the n variables. We call
the set of all n x 1 column matrices n-space and denote
it by 𝑹𝒏
 Sometimes we write an element of n-space as a
sequence of real numbers
𝒙 = (𝒙𝟏 , 𝒙𝟐 , . . . , 𝒙𝒏 )
called an ordered n-tuple.
𝑅𝑛 is an example of a vector space and we often refer to its elements as vectors. A
vector space is a non-empty set V with 2 operations (addition & scalar multiplication)
which have the properties for any u, v, w, in V and any real numbers k and m:
Section 4.2 in Textbook
Intro to Subspaces
 It is often the case that some vector space of interest is
contained within a larger vector space whose
properties are known.
 In this section we will show how to recognize when
this is the case, we will explain how the properties of
the larger vector space can be used to obtain
properties of the smaller vector space, and we will give
a variety of important examples.
Definition:
A subset W of vector space V is called a subspace of V if
W is itself a vector space under the addition and scalar
multiplication defined on V.
Theorem 4.2.1
If W is a set of one or more vectors in a vector space V
then W is a subspace of V if and only if the following
conditions are true:
If u and v are vectors in W then u+v is in W
b) If k is a scalar and u is a vector in W then ku is in W
a)
This theorem states that W is a subspace of V if and only
if it’s closed under addition and scalar multiplication.
Theorem 4.2.2:
Definition:
Theorem 4.2.3:
Example:
Section 4.3 in Textbook
Intro to Linear Independence
 In this section we will consider the question of whether the
vectors in a given set are interrelated in the sense that one
or more of them can be expressed as a linear combination
of others.
 In a rectangular xy-coordinate system every vector in the
plane can be expressed in exactly one way as a linear
combination of the standard unit vectors.
 Ex: express vector (3,2) as linear combination of 𝒊 = (1,0)
and 𝒋 = (0,1) is:
3,2 = 3 1,0 + 2 0,1 = 3𝒊 + 2𝒋
Theorem:
Note: span – a set of all linear combinations
For vectors 𝒗𝟏 , 𝒗𝟐 , . . . , 𝒗𝒓 in 𝑹𝒏 the following statements
are equivalent:
1) Any vector in the span of 𝒗𝟏 , 𝒗𝟐 , . . . , 𝒗𝒓 can be written
uniquely as a linear combination
2) If
then
3) None of the vectors 𝒗𝟏 , 𝒗𝟐 , . . . , 𝒗𝒓 is a linear
combination of the others.
Example:
Continued on Next Slide 
Example:
Example: Linear Independence in 𝑅
3
Continued on Next Slide 
Continued on Next Slide 
Example: Linear Independence in 𝑹𝟒
Section 4.4 in Textbook
Intro to Section 4.4
 We usually think of a line as being one-dimensional, a
plane as two-dimensional, and the space around us as
three-dimensional.
 It is the primary goal of this section and the next to
make this intuitive notion of dimension precise.
 In this section we will discuss coordinate systems in
general vector spaces and lay the groundwork for a
precise definition of dimension in the next section.
 In linear algebra coordinate systems are commonly
specified using vectors rather than coordinate axes. See
example below:
Units of Measurement
 They are essential ingredients of any coordinate
system. In geometry problems one tries to use the
same unit of measurement on all axes to avoid
distorting the shapes of figures. This is less important
in application
Questions to Get Done
Suggested practice problems (11th edition)
 Section 3.1 #1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23
 Section 3.2 #1, 3, 5, 7, 9, 11
 Section 3.3 #1, 13, 15, 17, 19
 Section 3.4 #17, 19, 25
Questions to Get Done
Suggested practice problems (11th edition)
 Section 4.2 #1, 7, 11
 Section 4.3 #3, 9, 11
 Section 4.4 #1, 7, 11, 13
 Section 4.5 #1, 3, 5, 13, 15, 17, 19
 Section 4.7 #1-19 (only odd)
 Section 4.8 #1, 3, 5, 7, 9, 15, 19, 21
Questions to Get Done
Suggested practice problems (11th edition)
 Section 6.2 #1, 7, 25, 27