Properties of subspace

Properties of subspace

Lessons

A subspace of Rn\Bbb{R}^n is any set SS in Rn\Bbb{R}^n that has the three following properties:
1) The zero vector is in SS
2) For each uu and vv in the set SS, the sum of u+vu+v is in SS (closed under addition)
3) For each uu in the set SS, the vector cucu is in SS. (closed under scalar multiplication)
  • 1.
    Properties of Subspace Overview:
    a)
    A Subspace in Rn \Bbb{R}^n
    • The zero vector
    • Closed under addition
    • Closed under scalar multiplication
    • Example of a Subspace
    • Example of not a Subspace

    b)
    Subspace of a span of vectors in Rn \Bbb{R}^n
    • Remember span = linear combination
    • Showing a span of vectors is a subspace in Rn\Bbb{R}^n


  • 2.
    Showing that a set is a subspace of Rn\Bbb{R}^n
    Is the following set a subspace of R2\Bbb{R}^2?
    Is this set a subspace of R^2

  • 3.
    Is the following set a subspace of R2\Bbb{R}^2?
    Is this set a subspace of R^2

  • 4.
    Showing that a set is a subspace of Rn\Bbb{R}^n with graphs
    The following graph displays a set in R2\Bbb{R}^2. Assume the set includes the bounding lines. Give a reason as to why the set SS is not a subspace of R2\Bbb{R}^2.
    graph of a set in R^2, graph 1

  • 5.
    Showing that a set is a subspace of Rn\Bbb{R}^n with graphs
    The following graph displays a set in R2\Bbb{R}^2. Assume the set includes the bounding lines. Give a reason as to why the set SS is not a subspace of R2\Bbb{R}^2.
    graph of a set in R^2, graph 2

  • 6.
    Showing a set equal to a span of vectors is a subspace of Rn\Bbb{R}^n
    Let U=U= Span{v1,v2,v3v_1,v_2,v_3}, where UU is a set. Determine if UU is in the subspace of R3\Bbb{R}^3.