Mathematical Immaturity

1.10 Exercises

In each of Exercises 1 through 10, let $S$ denote the set of all vectors $(x, y, z)$ in $V_3$ whose components satisfy the condition given. Determine whether $S$ is a subspace of $V_3.$ If $S$ is a subspace, compute $\dim S.$

• $x = 0.$
• $x + y = 0.$
• $x + y + z = 0.$
• $x = y.$
• $x = y = z.$
• $x = y$ or $x = z.$
• $x^2 - y^2 = 0.$
• $x + y = 1.$
• $y = 2x$ and $z = 3x.$
• $x + y + z = 0$ and $x - y - z = 0.$

Let $P_n$ denote the linear space of all real polynomials of degree $\leq n,$ where $n$ is fixed. In each of Exercises 11 through 20, let $S$ denote the set of all polynomials $f$ in $P_n$ satisfying the condition given. Determine whether or not $S$ is a subspace of $P_n.$ If $S$ is a subspace, compute $\dim S.$

• $f(0) = 0.$
• $f'(0) = 0.$
• $f''(0) = 0.$
• $f(0) + f'(0) = 0.$
• $f(0) = f(1).$
• $f(0) = f(2).$
• $f$ is even.
• $f$ is odd.
• $f$ has degree $\leq k,$ where $k < n,$ or $f = 0.$
• $f$ has degree $k,$ where $k < n,$ or $f = 0.$

• In the linear space of all real polynomials $p(t),$ describe the subspace spanned by each of the following subsets of polynomials and determine the dimension of this subspace:
  (a) $\{1, t^2, t^4\};$    (b) $\{t, t^3, t^5\};$    (c) $\{t, t^3\};$    (d) $\{1 + t, (1 + t)^2\}.$

• In this exercise, $L(S)$ denotes the subspace spanned by a subset $S$ of a linear space $V.$ Prove each of the statements (a) through (f).
  (a) $S \subseteq L(S).$
  (b) If $S \subseteq T \subseteq V$ and if $T$ is a subspace of $V,$ then $L(S) \subseteq T.$ This property is described by saying that $L(S)$ is the smallest subspace of $V$ which contains $S.$
  (c) A subset $S$ of $V$ is a subspace of $V$ if and only if $L(S) = S.$
  (d) If $S \subseteq T \subseteq V,$ then $L(S) \subseteq L(T).$
  (e) If $S$ and $T$ are subspaces of $V,$ then so is $S \cap T.$
  (f) If $S$ and $T$ are subsets of $V,$ then $L(S \cap T) \subseteq L(S) \cap L(T).$
  (g) Give an example in which $L(S \cap T) \neq L(S) \cap L(T).$

• 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.
  (a) $\{1, e^{ax}, e^{bx}\},$ $a \neq b.$
  (b) $\{e^{ax}, xe^{ax}\}.$
  (c) $\{1, e^{ax}, xe^{ax}\}.$
  (d) $\{e^{ax}, xe^{ax}, x^2e^{ax}\}.$
  (e) $\{e^x, e^{-x}, \cosh x\}.$
  (f) $\{\cos x, \sin x\}.$
  (g) $\{\cos^2 x, \sin^2 x\}.$
  (h) $\{1, \cos 2x, \sin^2 x\}.$
  (i) $\{\sin x, \sin 2x\}.$
  (j) $\{e^x \cos x, e^{-x} \sin x\}.$

• Let $V$ be a finite-dimensional linear space, and let $S$ be a subspace of $V.$ Prove each of the following statements.
  (a) $S$ is finite dimensional and $\dim S \leq \dim V.$
  (b) $\dim S = \dim V$ if and only if $S = V.$
  (c) Every basis for $S$ is part of a basis for $V.$
  (d) A basis for $V$ need not contain a basis for $S.$