Mathematical Immaturity

1.10 Exercises

• Let $V$ be the linear space consisting of all real-valued functions defined on the real line. Determine whether each of the following subsets of $V$ is dependent or independent. Compute the dimension of the subspace spanned by each set.
  $\quad\text{(a)}\quad$ $\{1, e^{ax}, e^{bx}\},$ $a \neq b.$
  $\quad\text{(b)}\quad$ $\{e^{ax}, xe^{ax}\}.$
  $\quad\text{(c)}\quad$ $\{1, e^{ax}, xe^{ax}\}.$
  $\quad\text{(d)}\quad$ $\{e^{ax}, xe^{ax}, x^2e^{ax}\}.$
  $\quad\text{(e)}\quad$ $\{e^x, e^{-x}, \cosh x\}.$
  $\quad\text{(f)}\quad$ $\{\cos x, \sin x\}.$
  $\quad\text{(g)}\quad$ $\{\cos^2 x, \sin^2 x\}.$
  $\quad\text{(h)}\quad$ $\{1, \cos 2x, \sin^2 x\}.$
  $\quad\text{(i)}\quad$ $\{\sin x, \sin 2x\}.$
  $\quad\text{(j)}\quad$ $\{e^x \cos x, e^{-x} \sin x\}.$
• Recall from Section 1.7 that a set $S$ of elements in a linear space $V$ is called dependent if there is a finite set of distinct elements in $S,$ say $x_1, \dots, x_k,$ and corresponding set of scalars $c_1, \dots, c_k,$ not all zero, such that \begin{align*} \sum_{i = 1}^k c_ix_i &= O. \end{align*} The set $S$ is called independent if it is not dependent. In this case, for all choices of distinct elements $x_1, \dots, x_k$ and scalars $c_1, \dots, c_k,$ \begin{align*} \sum_{i = 1}^k c_ix_i &= O \quad \text{implies} \quad c_1 = c_2 = \cdots = c_k = 0. \end{align*}
  
  From Section 1.8, we know that if a linear space $V$ has a basis with $n$ elements, then the integer $n$ is called the dimension of $V.$
• 
  $\text{(a)}\quad$ If $a = 0$ or $b = 0,$ then the subset $\{1, e^{ax}, e^{bx}\}$ is dependent. If $a \neq 0$ and $b \neq 0,$ then the subset is independent and the subspace spanned by the set has dimension $3.\quad\blacksquare$
  
  $\text{(b)}\quad$ The subset $\{e^{ax}, xe^{ax}\}$ is independent and the subspace spanned by the set has dimension $2.\quad\blacksquare$
  
  $\text{(c)}\quad$ If $a = 0,$ the subset $\{1, e^{ax}, xe^{ax}\}$ becomes $\{1, 1, x\},$ which is dependent. If $a \neq 0,$ the subset $\{1, e^{ax}, xe^{ax}\}$ is independent and the subspace spanned by the set has dimension $3.\quad\blacksquare$
  
  $\text{(d)}\quad$ The subset $\{e^{ax}, xe^{ax}, x^2e^{ax}\}$ is independent and the subspace spanned by the set has dimension $3.\quad\blacksquare$
  
  $\text{(e)}\quad$ Recall from Volume 1, Chapter 6, Section 18 that the hyperbolic cosine is given by the equation \begin{align*} \cosh &= \frac{e^x + e^{-x}}{2} \end{align*} In other words, the subset $\{e^x, e^{-x}, \cosh x\}$ can be rewritten as $$\left\{e^x, e^{-x}, \frac{e^x + e^{-x}}{2}\right\}$$ which is dependent. $\quad\blacksquare$
  
  $\text{(f)}\quad$ The subset $\{\cos x, \sin x\}$ is independent and the subspace spanned by the set has dimension $2.\quad\blacksquare$
  
  $\text{(g)}\quad$ The subset $\{\cos^2 x, \sin^2 x\}$ is independent and the subspace spanned by the set has dimension $2.\quad\blacksquare$
  
  $\text{(h)}\quad$ Recall the cosine's double-angle identity from Volume 1, Chapter 2, Section 5: \begin{align*} \cos 2x &= \cos^2 x - \sin^2 x = 1 - 2\sin^2 x \end{align*} From this, we can see that the subset $\{1, \cos 2x, \sin^2 x\}$ can be rewritten as \begin{align*} \left\{1, 1 - 2\sin^2 x, \sin^2 x\right\} \end{align*} which is a dependent set. $\quad \blacksquare$
  
  $\text{(i)}\quad$ Recall the sine's double-angle identity from Volume 1, Chapter 2, Section 5: \begin{align*} \sin 2x &= 2\sin x \cos x \end{align*} From this, we can see that the subset $\{1, \cos 2x, \sin^2 x\}$ can be rewritten as \begin{align*} \left\{\sin x, 2\sin x \cos x\right\} \end{align*} which is an independent set. The subspace spanned by this set has dimension $2. \quad \blacksquare$
  
  $\text{(j)}\quad$ The subset $\left\{e^x \cos x, e^{-x}\sin x\right\}$ is an independent set. The subspace spanned by this set has dimension $2. \quad \blacksquare$