Linear Algebra

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Linear Algebra

• Chapter 4 Real Vector Spaces

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Preliminaries

• At this point we have methods for solving linear
  systems, Gaussian Elimination and Cramers Rule,
  and a procedure for deciding if a solution exists
• For a square matrix, the central issue is
  singular versus nonsingular
• In these lectures we will generalize some
  concepts from two-dimensional and
  three-dimensional vectors to develop a deeper
  understanding of the solution process and the
  existence and uniqueness issues

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Chapter 4 Real Vector Spaces

• 4.1 Vectors in the Plane and in 3-Space
• 4.2 Vector Spaces
• 4.3 Subspaces
• 4.4 Span
• 4.5 Linear Independence
• 4.6 Basis and Dimension
• 4.7 Homogeneous Systems
• 4.8 Coordinates and Isomorphisms
• 4.9 Rank of a Matrix

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4.1 Vectors in the Plane and in 3-Space

• Defn - A vector in the plane is a 2 x 1 matrix
• where x and y are real numbers
• Notation - In print, use bold letters for
  vectors. When writing by hand use
• Defn - The numbers x and y in the definition of a
  vector are called the components of the vector v.
  
• Defn - Will say that two vectors and
  are equal if x1 x2 and y1 y2

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4.1 Vectors in the Plane and in 3-Space

• A two-dimensional vector has several
  geometric interpretations
• 1) A point (x, y) in the plane
• 2) A directed line segment from the origin to
  the point (x, y)
• 3) A directed line segment from the point (x1,
  y1) to the point (x2, y2). Then x x2 - x1, y
  y2 - y1
• For applications to geometry and physics, some of
  the most important operations are
• 1) Vector addition
• 2) Scalar multiplication
• 3) Vector subtraction

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4.1 Vectors in the Plane and in 3-Space

• Vector Addition
• Defn Let and . Define
• Geometric interpretation

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4.1 Vectors in the Plane and in 3-Space

• Scalar Multiplication
• Defn Let c be a scalar (e.g. a real number) and
  
• be a vector. The scalar multiple c u of u by c
  is
• defined as the vector

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4.1 Vectors in the Plane and in 3-Space

• Vector Subtraction
• Define u - v as u (-1) v
• Geometric relationship among u,v, u v and u - v
  

u (-v)
-v
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4.1 Vectors in the Plane and in 3-Space

• Basic Properties of Vectors in R2 or R3
• Theorem - If u, v and w are vectors in R2 or R3
  and c and d are real scalars, then
• a) u v v u
• b) u (v w) (u v) w
• c) u 0 0 u u
• d) u (-u) 0
• e) c (u v) c u c v
• f) (c d) u c u d u
• g) c (d u) (c d ) u
• h) 1 u u