Chapter 4: Real Vector Spaces

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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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4.4 Span

• Defn - Let S v1, v2, L, vk be a set of
  vectors in a vector space V. A vector v ? V is
  called a linear combination of the vectors in S
  if v a1 v1 a2 v2 L ak vk for some real
  numbers a1, a2 , K, ak

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4.4 Span

• Example
• Consider the three-dimensional vectors
  
• Express the vector as a linear
  combination of v1, v2 and v3

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4.4 Span

• Example (continued)
• The linear system may be solved to yield a1 1,
  a2 2
• and a3 -1. So, v v1 2 v2 - v3

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4.4 Span

• Defn - Let S v1, v2, K, vk be a set of
  vectors in a vector space V. The span of S is the
  set of all linear combinations of the elements of
  S
• To determine if a vector v belongs to the span of
  S, need to examine the corresponding system of
  linear equations. If that system has a solution
  (or solutions) then v belongs to the span of S

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4.4 Span

• Theorem - Let S v1, v2, K, vk be a set of
  vectors in a vector space V. The span of S is a
  subspace of V.
• Proof - To show that span(S) is a subspace of V,
  have to show
• If u, v Î span(S), then uv Î span(S)
• If c is a real number and u Î span(S), then cu Î
  span(S)
• Let u, v Î span(S). Then for
  some real numbers a1, a2, , ak and b1, b2, , bk
  
• 
  Î span(S).
• Let c be real and u Î span(S). Then for some
  real numbers a1, a2, , ak.

Î span(S).
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4.4 Span

• Defn - Let S v1, v2, K, vk be a set of
  vectors in a vector space V. The set S spans V if
  every vector in V is a linear combination of the
  vectors in S
• If span S V, then S is called a spanning set
  of V.

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4.4 Span

• Example
• Consider the homogeneous system Ax 0 where
  
• Determine the spanning set for the solution space
  to this homogeneous system.
• Form augmented matrix and put it into reduced row
  echelon form

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4.4 Span

• Example (continued)

Set x4 s ? x3 s Set x2 r ?
x1 - r - 2s