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Chapter 3: Q.13P (page 136)

In Problems $8 to 15, use (8.5)$ to show that the given functions are linearly independent.
$13. \sin 鈦� x, \sin 鈦� 2 x$

Short Answer

Expert verified

It has been shown that the functions $\sin 鈦� x, \sin 鈦� 2 x$ are linearly independent except at

$x = (n + \frac{1}{2}) 螤$

Step by step solution

01

Definition of linearly independent functions

The functions, $f_{1} (x), f_{2} (x), 鈥�, f_{n} (x)$ are linearly independent if the determinant

$W = |\begin{array}{cccc} f_{1} (x) & f_{2} (x) & 鈰� & f_{n} (x) \\ f_{1}^{鈥�} (x) & f_{2}^{鈥�} (x) & 鈰� & f_{n}^{鈥�} (x) \\ 鈰� & 鈰� & 鈰� & 鈰� \\ f_{1}^{(n 鈭� 1)} (x) & f_{2}^{(n 鈭� 1)} (x) & 鈰� & f_{n}^{(n 鈭� 1)} (x) \end{array}| 猝� 0 鈭�$

Here, W is called the Wronksian of functions.

02

Use the Wronksian to show that the given functions are linearly independent.

Find the derivatives of the function of order $f_{1} (x) = \sin 鈦� x of order 1$

$\begin{aligned} f_{1} (x) = \sin 鈦� x \\ f_{1}^{鈥�} (x) = \cos 鈦� x \end{aligned}$

Find the derivatives of the function of order $f_{2} (x) = \sin 鈦� 2 x of order 1$

$\begin{aligned} f_{2} (x) = \sin 鈦� 2 x \\ f_{2}^{鈥�} (x) = 2 \cos 鈦� 2 x \end{aligned}$

Substitute the derivatives in the Wronksian formula and simplify as follows:

localid="1657426787997" $\begin{aligned} W = |\begin{array}{cc} \sin 鈦� x & \sin 鈦� 2 x \\ \cos 鈦� x & 2 \cos 鈦� 2 x \end{array}| \\ = 2 \sin 鈦� x \cos 鈦� 2 x 鈭� \sin 鈦� 2 x \cos 鈦� x \end{aligned}$

Use the trigonometric identities $\cos 鈦� 2 x = \cos^{2} 鈦� x 鈭� \sin^{2} 鈦� x and \sin 鈦� 2 x = 2 \sin 鈦� x \cos 鈦� x$

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Most popular questions from this chapter

Let each of the following matrices M describe a deformation of the $(x, y)$ plane for each given Mfind: the Eigen values and eigenvectors of the transformation, the matrix Cwhich Diagonalizesand specifies the rotation to new axesrole="math" localid="1658833126295" $(x', y')$ along the eigenvectors, and the matrix D which gives the deformation relative to the new axes. Describe the deformation relative to the new axes.

role="math" localid="1658833142584" $(\begin{matrix} 3 & 1 \\ 1 & 3 \end{matrix})$

Show that the given lines intersect and find the acute angle between them.

$r = 2 j + k + (3 i - k) t_{1} and r = 7 i + 2 k + (2 i - j + k) t_{2}$

Let each of the following matrices represent an active transformation of vectors in (x,y)plane (axes fixed, vector rotated or reflected). As in Example 3, show that each matrix is orthogonal, find its determinant and find its rotation angle, or find the line of reflection. $\frac{1}{5} (\begin{matrix} 3 & 4 \\ 4 & - 3 \end{matrix})$ $\frac{1}{5} (\begin{matrix} 3 & 4 \\ 4 & - 3 \end{matrix})$

Show that the following matrices are Hermitian whether Ais Hermitian or not: $A A^{t}, A + A^{t}, i (A - A^{t})$ .

Show that each of the following matrices is orthogonal and find the rotation and/or reflection it produces as an operator acting on vectors. If a rotation, find the axis and angle; if a reflection, find the reflecting plane and the rotation, if any, about the normal to that plane.

$M = \frac{1}{2} (\begin{matrix} 1 & \sqrt{2} & - 1 \\ \sqrt{2} & 0 & \sqrt{2} \\ 1 & - \sqrt{2} & - 1 \end{matrix})$

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