Chapter 6: Q12E (page 326)

Determine whether the given functions are linearly dependent or linearly independent on the specified interval. Justify your decisions.
${\cos 2 x, \cos^{2} x, \sin^{2} x}$ on $(- \infty, \infty)$

Short Answer

Expert verified

Therefore, the function $\{\cos 2 x, \cos^{2} x, \sin^{2} x\}$ is linearly dependent on $(- \infty, \infty)$ .

Step by step solution

Using the concept of Wronskian

The given function is $\{\cos 2 x, \cos^{2} x, \sin^{2} x\} .$

Apply the concept of Wronskian,

$W [f_{1}, f_{2}, \dots, f_{n}] = |\begin{matrix} f_{1} (x) & f_{2} (x) & \dots & f_{n} (x) \\ f_{1}' (x) & f_{2}' (x) & \dots & f_{n}' (x) \\ ⋮ & ⋮ & ⋮ \\ {f_{1}}^{n - 1} (x) & {f_{2}}^{n - 1} (x) & \dots & {f_{n}}^{n - 1} (x) \end{matrix}|$

Therefore,

$W [\cos 2 x, \cos^{2} x, \sin^{2} x] = W [\cos 2 x, \frac{1 + \cos 2 x}{2}, \frac{1 - \cos 2 x}{2}] = |\begin{matrix} \cos 2 x & \frac{1 + \cos 2 x}{2} & \frac{1 - \cos 2 x}{2} \\ - 2 \sin 2 x & - \sin 2 x & \sin 2 x \\ - 4 \cos 2 x & - 2 \cos 2 x & 2 \cos 2 x \end{matrix}|$

Solve the above equation,

$W [\cos 2 x, \cos^{2} x, \sin^{2} x] = \cos 2 x (- 2 \sin 2 xcos 2 x + 2 \sin 2 xcos 2 x) - \frac{1 + \cos 2 x}{2} (- 4 \sin 2 xcos 2 x + 4 \sin 2 xcos 2 x) + \frac{1 - \cos 2 x}{2} (4 \sin 2 xcos 2 x - 4 \sin 2 xcos 2 x) = \cos 2 x (0) - \frac{1 + \cos 2 x}{2} (0) + \frac{1 - \cos 2 x}{2} (0) = 0$

Step 2:Check the linearly independent or dependent

The above function is equal to zero $(\forall x) .$

Therefore, $\{\cos 2 x, \cos^{2} x, \sin^{2} x\}$ is linearly dependent on $(- \infty, \infty)$ .

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Determine whether the given functions are linearly dependent or linearly independent on the specified interval. Justify your decisions.
${\cos 2 x, \cos^{2} x, \sin^{2} x}$ on $(- \infty, \infty)$

Short Answer

Step by step solution

Using the concept of Wronskian

Step 2:Check the linearly independent or dependent

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Determine whether the given functions are linearly dependent or linearly independent on the specified interval. Justify your decisions.{cos2x, cos2x, sin2x}on(-∞, ∞)

Short Answer

Step by step solution

Using the concept of Wronskian

Step 2:Check the linearly independent or dependent

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Theoretical and Mathematical Physics

Applied Mathematics

Pure Maths

Decision Maths

Statistics

Study anywhere. Anytime. Across all devices.

Determine whether the given functions are linearly dependent or linearly independent on the specified interval. Justify your decisions.
${\cos 2 x, \cos^{2} x, \sin^{2} x}$ on $(- \infty, \infty)$