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Slide 1
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Introduction
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Continued on Next Slide
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Section 3.1 in Textbook
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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.
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Section 4.2 in Textbook
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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.
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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.
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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: a) 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 This
theorem states that W is a subspace of V if and only if its closed
under addition and scalar multiplication.
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Theorem 4.2.2: Definition:
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Theorem 4.2.3:
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Example:
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Section 4.3 in Textbook
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Intro to Linear Independence
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Theorem:
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Example: Continued on Next Slide
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Example:
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Continued on Next Slide
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Section 4.4 in Textbook
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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.
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In linear algebra coordinate systems are commonly specified
using vectors rather than coordinate axes. See example below:
Slide 39
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