- Calculus, Volume 2: Multi-Variable Calculus and Linear Algebra with Applications to Differential Equations and Probability
- Tom M. Apostol
- Second Edition
- 1991
- 978-1-119-49676-2
1.17 Exercises
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In each case, find an orthonormal basis for the subspace of $V_3$ spanned by the given vectors.
$\qquad \text{(a)}\quad x_1 = (1, 1, 1), \quad x_2 = (1, 0, 1), \quad x_3 = (3, 2, 3).$
$\qquad \text{(b)}\quad x_1 = (1, 1, 1), \quad x_2 = (-1, -1, -1), \quad x_3 = (1, 0, 1).$
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In each case, find an orthonormal basis for the subspace of $V_4$ spanned by the given vectors.
$\qquad \text{(a)}\quad x_1 = (1, 1, 0, 0), \quad x_2 = (0, 1, 1, 0), \quad x_3 = (0, 0, 1, 1), \quad x_4 = (1, 0, 0, 1).$
$\qquad \text{(b)}\quad x_1 = (1, 1, 0, 1), \quad x_2 = (1, 0, 2, 1), \quad x_3 = (1, 2, -2, 1).$
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In the real linear space $C(0, \pi),$ with inner product $(x, y) = \int_0^\pi x(t) y(t) \, dt,$ let $x_n(t) = \cos nt$ for $n = 0, 1, 2, \dots.$ Prove that the functions $y_0, y_1, y_2, \dots,$ given by
\begin{align*}
y_0(t) = \frac{1}{\sqrt{\pi}}, \quad y_n(t) = \frac{\sqrt{2}}{\sqrt{\pi}} \cos nt \quad \text{for} \quad n \geq 1,
\end{align*}
form an orthonormal set spanning the same subspace as $x_0, x_1, x_2, \dots.$
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In the linear space of all real polynomials, with inner product $(x, y) = \int_0^1 x(t) y(t) \, dt,$ let $x_n(t) = t^n$ for $n = 0, 1, 2, \dots.$ Prove that the functions
\begin{align*}
y_0(t) = 1, \quad y_1(t) = \sqrt{3} (2t - 1), \quad y_n(t) = \sqrt{5} (6t^2 - 6t + 1)
\end{align*}
form an orthonormal set spanning the same subspace as $\{x_0, x_1, x_2, \dots\}.$
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Let $V$ be the linear space of all real functions $f$ continuous on $[0, +\infty)$ and such that the integral $\int_0^\infty e^{-t} f^2(t) \, dt$ converges. Define $(f, g) = \int_0^\infty e^{-t} f(t) g(t) \, dt,$ and let $y_0, y_1, y_2, \dots,$ be the set obtained by applying the Gram-Schmidt process to $x_0, x_1, x_2, \dots,$ where $x_n(t) = t^n$ for $n \geq 0.$ Prove that $y_0(t) = 1,$ $y_1(t) = t - 1,$ $y_2(t) = t^2 - 4t + 2,$ $y_3(t) = t^3 - 9t^2 + 18t - 6.$