Homework for Span

1. When is a vector an element of a span?

Given a finite subset $S$ of a vector space $V,$ and a single vector ${\bf v} \in V,$ know how to determine when ${\bf v} \in \textrm{Span}(S).$
Note: Showing that ${\bf v} \in \textrm{Span}(S)$ will require actually expressing ${\bf v}$ as a particular linear combination of the vectors in $S.$

Examples:
- From http://linear.pugetsound.edu/html/section-S.html, read Example LCM, Example SSP, and Example SM32.
  Note: Beezer uses the notation $\langle S \rangle$ for $\textrm{Span}(S).$ We will use the latter.
- Example: Let $\mathcal{F}_2 = \left\{ \cos(2t) , \sin(2t), 1 \right\} \subseteq C^{\ \infty}(\mathbb{R}).$ Use the power reduction formulas to show $\cos^2(t) \in \textrm{Span}(\mathcal{F}_2).$
  Solution: The power reduction formula for $\cos^2(t)$ gives us $$\cos^2(t) = \frac{1 + \cos(2t)}{2} = \frac{1}{2} (1) + \frac{1}{2} (\cos(2t))$$ which is a linear combination of the vectors $1$ and $\cos(2t) \in \mathcal{F}_2,$ thus $\cos^2(t) \in \textrm{Span}(\mathcal{F}_2).$
Homework:
- From http://linear.pugetsound.edu/html/section-S.html, do Exercises S.C15, S.C16, S.C17, S.C20, and S.C21.
  Note: For Exercises S.C16, S.C17, and S.C21, express the vector ${\bf v}$ as a linear combination of vectors from the set $S.$
- Find the appropriate formulas here:
  - https://en.wikipedia.org/wiki/List_of_trigonometric_identities
  - https://en.wikipedia.org/wiki/Taylor_series
  1. Let $\mathcal{F}_2 = \left\{ \cos(2t) , \sin(2t), 1 \right\}$ and $\mathcal{T}_2 = \left\{ \cos^2(t) , \cos(t) \sin(t), \sin^2(t) \right\}.$ Use the power reduction formulas and the double angle identities to show the following:
    1. Show $\mathcal{T}_2 \subseteq \textrm{Span}(\mathcal{F}_2).$
    2. Show $\mathcal{F}_2 \subseteq \textrm{Span}(\mathcal{T}_2).$
    Note: These two facts along with Theorem 2 show that $\textrm{Span}(\mathcal{T}_2) = \textrm{Span}(\mathcal{F}_2).$
  2. Let $\mathcal{F}_3 = \left\{ \cos(t) , \sin(t) , \cos(3t) , \sin(3t) \right\}$ and $\mathcal{T}_3 = \left\{ \cos^3(t) , \cos^2(t) \sin(t) , \cos(t) \sin^2(t) , \sin^3(t) \right\}.$ Use the power reduction formulas and the triple angle identities to show the following:
    1. Show $\mathcal{T}_3 \subseteq \textrm{Span}(\mathcal{F}_3).$
    2. Show $\mathcal{F}_3 \subseteq \textrm{Span}(\mathcal{T}_3).$
    Note: These two facts along with Theorem 2 show that $\textrm{Span}(\mathcal{T}_3) = \textrm{Span}(\mathcal{F}_3).$
  3. Let $\mathcal{M}_0 = \left\{ 1 , t , t^2 , t^3 \right\}$ and $\mathcal{M}_1 = \left\{ 1 , (t-1) , (t-1)^2 , (t-1)^3 \right\}.$ Use the Taylor Series Formula to show the following:
    1. Show $\mathcal{M}_1 \subseteq \textrm{Span}(\mathcal{M}_0).$
    2. Show $\mathcal{M}_0 \subseteq \textrm{Span}(\mathcal{M}_1).$
    Note: These two facts along with Theorem 2 show that $\textrm{Span}(\mathcal{M}_1) = \textrm{Span}(\mathcal{M}_0) = \mathcal{P}_3.$

2. Find a spanning set for $V.$

Given a vector space $V,$ find a finite subset $S \subseteq V$ that spans $V.$

Examples:
- Example: Let $V = \left\{ p \in \mathcal{P}_3 \ \middle\vert \ p(2) = p(0) \textrm{ and } p^{\ \prime}(1) = 0 \right\}.$ Find a spanning set $S$ for $V$.
  Solution: If $p \in \mathcal{P}_3,$ we can express $p$ as $$p(t) = a_3 t^3 + a_2 t^2 + a_1 t + a_0.$$ Then $$\begin{array}{rcl} p(2) & = & 8 a_3 + 4 a_2 + 2 a_1 + a_0 = p(0) = a_0 \\ \Rightarrow & & 8 a_3 + 4 a_2 + 2 a_1 = 0 \\ \Rightarrow & & 4 a_3 + 2 a_2 + a_1 = 0 \\ \end{array}$$ and $$p^{\ \prime}(1) = 3 a_3 + 2 a_2 + a_1 = 0$$ gives us the homogeneous linear system of equations $$\begin{array}{rcl} 4 a_3 + 2 a_2 + a_1 & = & 0 \\ 3 a_3 + 2 a_2 + a_1 & = & 0 \\ \end{array}$$ whose augmented matrix reduces to $$\left[ \begin{array}{cccc} 4 & 2 & 1 & 0 \\ 3 & 2 & 1 & 0 \\ \end{array} \right] \rightarrow \left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & \frac{1}{2} & 0 \\ \end{array} \right]$$ $\Rightarrow p(t) = a_3 t^3 + a_2 t^2 + a_1 t + a_0$ has coefficients satisfying the equations $a_3 = 0$ and $a_2 = -\frac{1}{2} a_1.$ Thus $$\begin{array}{rcl} p(t) & = & a_3 t^3 + a_2 t^2 + a_1 t + a_0 \\ & = & 0 \cdot t^3 -\frac{1}{2} a_1 t^2 + a_1 t + a_0 \\ & = & a_1 \left(-\frac{1}{2} t^2 + t \right) + a_0 (1) \\ \end{array}$$ So $$V = \left\{ \ a_1 \left( -\frac{1}{2} t^2 + t \right) + a_0 (1) \ \middle\vert \ a_1 , a_0 \in \mathbb{R} \ \right\} = \textrm{Span}\left\{ -\frac{1}{2} t^2 + t \ , \ 1 \right\}.$$
Homework: For each of the following, find a spanning set $S.$
- 1. $V = \left\{ y \ \middle\vert \ y^{\ \prime \prime} - 4 y = 0 \textrm{ and } y(0) = 0 \ \right\}.$
  2. $V = \left\{ y \ \middle\vert \ y^{\ \prime \prime} - 4 y = 0 \textrm{ and } y^{\ \prime}(0) = 0 \ \right\}.$
  3. $V = \left\{ y \ \middle\vert \ y^{\ \prime \prime} + y = 0 \textrm{ and } y \left(\frac{\pi}{4}\right) = 0 \ \right\}.$
  4. $V = \left\{ p \in \mathcal{P}_5 \ \middle\vert \ p(1) = 0 \textrm{ and } p(-1) = 0 \right\}.$
  5. $V = \left\{ p \in \mathcal{P}_2 \ \middle\vert \ p^{\ \prime}(1) = 0 \right\}.$
  6. $V = \left\{ p \in \mathcal{P}_4 \ \middle\vert \ p^{\ \prime}(1) = 0 \textrm{ and } p(-2) = 0 \right\}.$
  7. $$V = \left\{ A \in M_2(\mathbb{R}) \ \middle\vert \ A \left[ \begin{array}{c} 1 \\ -2 \\ \end{array} \right] = \vec{0} \right\}.$$
  8. $$V = \left\{ A \in M_3(\mathbb{R}) \ \middle\vert \ A \left[ \begin{array}{cc} 1 & 1 \\ -1 & 0 \\ 0 & -1 \\ \end{array} \right] = \left[ \begin{array}{cc} \vec{0} & \vec{0} \\ \end{array} \right] \right\}.$$