Chapter 3 4 Euclidean General Vector Spaces MATH

Chapter 3 - 4 = Euclidean & General Vector Spaces MATH 264 Linear Algebra

Introduction

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Vectors in 2 -space, 3 space, and n-space 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.

Subspaces 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 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:

Linear Independence Section 4. 3 in Textbook

Intro to Linear Independence

Theorem:

Example: Continued on Next Slide

Example:

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Coordinates & Basis Section 4. 4 in Textbook

Intro to Section 4. 4 Ø We usually think of a line as being one-dimensional, a plane as two-dimensional, and the space around us as three-dimensional. Ø It is the primary goal of this section and the next to make this intuitive notion of dimension precise. Ø 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 1 ( 1 th 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 1 ( 1 th 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 1 ( 1 th edition) Section 6. 2 #1, 7, 25, 27