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Vector Subspaces

Module by: Mark A. Davenport. E-mail the author

Defintition 1

A (non-empty) subset WW of VV is called a supspace of VV if for any x,y∈Wx,y∈W, span ({x,y})⊆W span ({x,y})⊆W.

Note that this definition easily implies that:

  • 0 ∈ W 0 ∈ W
  • WW is itself a vector space

Example 1: Which of these are subspaces?

  •   An illustration of an ellipse-shaped set W.   [No]
  •   An illustration of a line passing through the origin.   [Yes]
  • V=R5V=R5, W={x:x4=0,x5=0}W={x:x4=0,x5=0} [Yes]
  • V=R5V=R5, W={x:x4=1,x5=1}W={x:x4=1,x5=1} [No]
  • V=C[0,1]V=C[0,1], W=W={polynomials of degree NN} [Yes]
  • V=C[0,1]V=C[0,1], W={f:fW={f:f is bandlimited to BB} [Yes]
  • V=RNV=RN, W={x:x hasnomorethan 5 nonzerocomponents ,i.e.,x0≤5}W={x:x hasnomorethan 5 nonzerocomponents ,i.e.,x0≤5} [No]

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