Vector Spaces MATH 264 Linear Algebra Introduction There are two types of physical quantities Scalars quantities that can be described by numerical value alone Ex temperature length speed ID: 655137
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Chapter 3 - 4 =Euclidean & GeneralVector Spaces
MATH
264 Linear
AlgebraSlide2
IntroductionThere 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
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Continued on Next Slide
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Vectors in 2-space, 3-space, and n-spaceSection 3.1 in TextbookSlide5
DefinitionsVectors 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.Slide6
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 Sometimes we write an element of n-space as a sequence of real numbers
called an
ordered n-tuple.
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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: Slide8Slide9Slide10
SubspacesSection 4.2 in TextbookSlide11
Intro to SubspacesIt 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.Slide12
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.Slide13Slide14
Theorem 4.2.1If 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
If
k is a scalar and u
is a vector in
W
then
ku
is in
W
This theorem states that
W
is a subspace of
V
if and only if it’s closed under addition and scalar multiplication.Slide15Slide16Slide17Slide18
Theorem 4.2.2:Definition:Slide19
Theorem 4.2.3:Slide20Slide21
Example:Slide22
Linear IndependenceSection 4.3 in TextbookSlide23
Intro to Linear IndependenceIn 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
and
is:
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Theorem:Note: span – a set of all linear combinationsFor vectors
in
the following statements are equivalent:
Any vector in the span of
can be written uniquely as a linear combination
If then
None of the vectors
is a linear combination of the others.
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Example:Continued on Next Slide Slide28Slide29Slide30
Example:Slide31
Example: Linear Independence in
Continued on Next Slide
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Continued on Next Slide Slide33Slide34
Example: Linear Independence in Slide35
Coordinates & BasisSection 4.4 in TextbookSlide36
Intro to Section 4.4We 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.Slide37Slide38
In linear algebra coordinate systems are commonly specified using vectors rather than coordinate axes. See example below:Slide39
Units of MeasurementThey 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 applicationSlide40Slide41Slide42Slide43Slide44Slide45Slide46Slide47Slide48
Questions to Get DoneSuggested practice problems (11th edition) Section 3.1 #1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23Section 3.2 #1, 3, 5, 7, 9, 11Section 3.3 #1, 13, 15, 17, 19Section 3.4 #17, 19, 25Slide49
Questions to Get DoneSuggested practice problems (11th edition) Section 4.2 #1, 7, 11Section 4.3 #3, 9, 11Section 4.4 #1, 7, 11, 13Section 4.5 #1, 3, 5, 13, 15, 17, 19Section 4.7 #1-19 (only odd)Section 4.8 #1, 3, 5, 7, 9, 15, 19, 21Slide50
Questions to Get DoneSuggested practice problems (11th edition) Section 6.2 #1, 7, 25, 27