Homework for Subspaces

1. Subspaces

• The definition of a subspace
• The three properties we check to see if a subset is a subspace
• Examples of subspaces

• Examples:
  • From our text, Differential Equations and Their Applications by Martin Braun, section 3.2, read Examples 1, 2, and 7.
• Homework:
  • From http://linear.pugetsound.edu/html/section-S.html, do Reading Questions 1 and 2 and Exercises S.C25, S.C26, S.M20, S.T20, S.T30, and S.T31.
  • From our text, Differential Equations and Their Applications by Martin Braun, section 3.2, do Exercises 1, 2, 3, 4, 5, 6, 9, 10, 11, and 12.
    Note:
    • For the Exercises in which the subset is a subspace, do not prove all 8 of the vector space axioms. Instead, prove that the set in question satisfies Property Z, Property AC, and Property SC to prove that it is a subspace.
    • Regarding the Exercises in which the subset is not a subspace: in each case, it is clear immediately that the subset fails Property Z, but it is a useful exercise to create counterexamples in each case showing that the subset also fails Property AC and Property SC.

2. Null Space

• Examples:
  • From http://linear.pugetsound.edu/html/section-HSE.html, read Example CNS1 and Example CNS2.
    Note:
    • Beezer uses the notation $\mathcal{N}(A)$ for $\textrm{Nul}(A).$ We will continute to use the latter.
    • We will continue to use parametric form for our solutions when describing $\textrm{Nul}(A).$ For example, our solution to Example CNS1 would be $$\textrm{Nul}(A) = \left\{ \ x_3 \left[\begin{array}{c} -2 \\ 3 \\ 1 \\ 0 \\ 0 \\ \end{array}\right] + x_5 \, \left[\begin{array}{c} -1 \\ -4 \\ 0 \\ -2 \\ 1 \\ \end{array}\right] \ \middle\vert \ x_3 , x_5 \in \mathbb{R} \ \right\}.$$
• Homework:
  • From http://linear.pugetsound.edu/html/section-HSE.html, do Exercises HSE.C30 and HSE.C31. Put your solutions in parametric form.