Chapter 3 - 4 = Euclidean & General Vector Spaces

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

2. 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

3. Continued on Next Slide 

4. Vectors in 2-space, 3-space, and n-space Section 3.1 in Textbook

5. 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.

6. 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.

7. 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:

10. Subspaces Section 4.2 in Textbook

11. 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.

12. 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.

14. 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 • 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.

18. Theorem 4.2.2: Definition:

19. Theorem 4.2.3:

21. Example:

22. Linear Independence Section 4.3 in Textbook

23. 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 and is:

25. Theorem: Note: span – a set of all linear combinations For 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.

27. Example: Continued on Next Slide 

30. Example:

31. Example: Linear Independence in Continued on Next Slide 

32. Continued on Next Slide 

34. Example: Linear Independence in

35. Coordinates & Basis Section 4.4 in Textbook

36. 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.

38. In linear algebra coordinate systems are commonly specified using vectors rather than coordinate axes. See example below:

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

48. 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

49. 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

50. Questions to Get Done Suggested practice problems (11th edition) • Section 6.2 #1, 7, 25, 27