Chapter 6: Q8E (page 326)

Determine whether the given functions are linearly dependent or linearly independent on the specified interval. Justify your decisions.
${x^{2}, x^{2} - 1, 5}$ on $(- \infty, \infty)$

Short Answer

Expert verified

Thus, $\{x^{2}, x^{2} - 1, 5\}$ are linearly dependent on $(- \infty, \infty)$ .

Step by step solution

Step 1:Using the concept of Wronskian

The given function is $\{x^{2}, x^{2} - 1, 5\} .$

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,

$W [x^{2}, x^{2} - 1, 5] = |\begin{matrix} x^{2} & x^{2} - 1 & 5 \\ 2 x & 2 x & 0 \\ 2 & 2 & 0 \end{matrix}|$

Solve the above equation,

$W [x^{2}, x^{2} - 1, 5] = x^{2} (0) - (x^{2} - 1) (0) + 5 (4 x - 4 x) = 0$

Step 2:Check the linearly independent or dependent

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

Therefore, $\{x^{2}, x^{2} - 1, 5\}$ are 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.
${x^{2}, x^{2} - 1, 5}$ on $(- \infty, \infty)$

Short Answer

Step by step solution

Step 1: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.{x2, x2-1, 5}on(-∞, ∞)

Short Answer

Step by step solution

Step 1: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

Theoretical and Mathematical Physics

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Study anywhere. Anytime. Across all devices.

Determine whether the given functions are linearly dependent or linearly independent on the specified interval. Justify your decisions.
${x^{2}, x^{2} - 1, 5}$ on $(- \infty, \infty)$