Chapter 6: Q11E (page 326)

Determine whether the given functions are linearly dependent or linearly independent on the specified interval. Justify your decisions.
${x^{- 1}, x^{\frac{1}{2}}, x}$ on $(0, \infty)$

Short Answer

Expert verified

Therefore, $\{x^{- 1}, x^{\frac{1}{2}}, x\}$ are linearly independent on $(0, \infty)$ .

Step by step solution

Using the concept of Wronskian

The given function is $\{x^{- 1}, x^{\frac{1}{2}}, x\} .$

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,

$W [x^{- 1}, x^{\frac{1}{2}}, x] = |\begin{matrix} x^{- 1} & x^{\frac{1}{2}} & x \\ - x^{- 2} & \frac{1}{2} x^{- \frac{1}{2}} & 1 \\ 2 x^{- 3} & - \frac{1}{4} x^{- \frac{3}{2}} & 0 \end{matrix}|$

Solve the above equation,

$W [x^{- 1}, x^{\frac{1}{2}}, x] = x^{- 1} (0 + \frac{1}{4} x^{- \frac{3}{2}}) - x^{\frac{1}{2}} (0 - 2 x^{- 3}) + x (\frac{1}{4} x^{- 2 - \frac{3}{2}} - x^{- 3 - \frac{1}{2}}) = \frac{1}{4} (x^{- \frac{5}{2}}) + (2 x^{\frac{- 5}{2}}) + (\frac{1}{4} x^{1 - \frac{7}{2}} - x^{1 - \frac{7}{8}}) = \frac{9}{4} (x^{- \frac{5}{2}}) - \frac{3}{4} x^{- \frac{5}{2}} = \frac{6}{4} (x^{- \frac{5}{2}}) = \frac{3}{2} (x^{- \frac{5}{2}})$

Step 2:Check the linearly independent or dependent

The above function is not equal to zero $(\forall x)$

Therefore, $\{x^{- 1}, x^{\frac{1}{2}}, x\}$ are linearly independent on $(0, \infty)$ .

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Determine whether the given functions are linearly dependent or linearly independent on the specified interval. Justify your decisions.
${x^{- 1}, x^{\frac{1}{2}}, x}$ on $(0, \infty)$

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.{x-1, x12, x}on(0, ∞)

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

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Theoretical and Mathematical Physics

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Determine whether the given functions are linearly dependent or linearly independent on the specified interval. Justify your decisions.
${x^{- 1}, x^{\frac{1}{2}}, x}$ on $(0, \infty)$