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Chapter 4: Problem 91

Show that the following functions \(f, g, h\) are linearly independent: (a) \(f(t)=e^{t}, g(t)=\sin t, h(t)=t^{2}\); (b) \(f(t)=e^{t}, g(t)=e^{2 t}, h(t)=t\).

Short Answer

Expert verified
The Wronskian determinants for both cases are found to be non-zero: Case 1: \( W(f,g,h) = e^t(2\cos t + 2t\sin t) - \sin t(2e^t - 2te^t) + t^2(e^t\sin t - e^t\cos t) \) Case 2: \( W(f,g,h) = -4e^{3t} + e^{3t} + t2e^{3t} \) Hence, the functions in both cases (a) \(f(t)=e^{t}, g(t)=\sin{t}, h(t)=t^2\) and (b) \(f(t)=e^{t}, g(t)=e^{2t}, h(t)=t\) are linearly independent.

Step by step solution

01

Find the derivatives of f(t), g(t), and h(t)

First, we need to determine the first and the second derivative of each function. For \(f(t)=e^{t}\): \(f'(t)=e^{t}\) and \(f''(t)=e^{t}\). For \(g(t)=\sin{t}\): \(g'(t)=\cos{t}\) and \(g''(t)=-\sin{t}\). For \(h(t)=t^{2}\): \(h'(t)=2t\) and \(h''(t)=2\).
02

Form the Wronskian determinant

Now, let's form the Wronskian determinant with the functions and their derivatives: \( W(f,g,h) = \begin{vmatrix} e^t & \sin t & t^2 \\ e^t & \cos t & 2t \\ e^t & -\sin t & 2 \end{vmatrix} \)
03

Calculate the Wronskian determinant

Let's evaluate the determinant: \( W(f,g,h) = e^t \begin{vmatrix} \cos t & 2t \\ -\sin t & 2 \\ \end{vmatrix} - \sin t \begin{vmatrix} e^t & 2t \\ e^t & 2 \\ \end{vmatrix} + t^2 \begin{vmatrix} e^t & \cos t \\ e^t & -\sin t \\ \end{vmatrix} \) \( W(f,g,h) = e^t(2\cos t + 2t\sin t) - \sin t(2e^t - 2te^t) + t^2(e^t\sin t - e^t\cos t) \) This Wronskian determinant is non-zero. Thus, the functions \(f(t)=e^{t}\), \(g(t)=\sin{t}\), and \(h(t)=t^2\) are linearly independent. #Case 2: f(t)=e^t, g(t)=e^{2t}, h(t)=t#
04

Find the derivatives of f(t), g(t), and h(t)

First, we need to determine the first and the second derivative of each function. For \(f(t)=e^{t}\): \(f'(t)=e^{t}\) and \(f''(t)=e^{t}\). For \(g(t)=e^{2t}\): \(g'(t)=2e^{2t}\) and \(g''(t)=4e^{2t}\). For \(h(t)=t\): \(h'(t)=1\) and \(h''(t)=0\).
05

Form the Wronskian determinant

Now, let's form the Wronskian determinant with the functions and their derivatives: \( W(f, g, h) = \begin{vmatrix} e^t & e^{2t} & t \\ e^t & 2e^{2t} & 1 \\ e^t & 4e^{2t} & 0 \end{vmatrix} \)
06

Calculate the Wronskian determinant

Let's evaluate the determinant: \( W(f,g,h) = e^t \begin{vmatrix} 2e^{2t} & 1 \\ 4e^{2t} & 0 \\ \end{vmatrix} - e^{2t} \begin{vmatrix} e^t & 1 \\ e^t & 0 \\ \end{vmatrix} + t \begin{vmatrix} e^t & 2e^{2t} \\ e^t & 4e^{2t} \\ \end{vmatrix} \) \( W(f,g,h) = e^t(0 - 4e^{2t}) - e^{2t}(0 - e^t) + t(e^t4e^{2t} - e^t2e^{2t}) \) \( W(f,g,h) = -4e^{3t} + e^{3t} + t2e^{3t} \) This Wronskian determinant is non-zero. Thus, the functions \(f(t)=e^{t}\), \(g(t)=e^{2t}\), and \(h(t)=t\) are linearly independent.

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Most popular questions from this chapter

Find the value of \(k\) that satisfies the following equation: $$ \operatorname{det}\left(\begin{array}{ccc} 2 a_{1} & 2 a_{2} & 2 a_{3} \\ 3 b_{1}+5 c_{1} & 3 b_{2}+5 c_{2} & 3 b_{3}+5 c_{3} \\ 7 c_{1} & 7 c_{2} & 7 c_{3} \end{array}\right)=k \operatorname{det}\left(\begin{array}{lll} a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{array}\right) . $$

Let \(y_{1}, y_{2}, \ldots, y_{n}\) be linearly independent functions in \(C^{\infty}\). For each \(y \in \mathrm{C}^{\infty}\), define $\mathrm{T}(y) \in \mathrm{C}^{\infty}$ by $$ [\mathrm{T}(y)](t)=\operatorname{det}\left(\begin{array}{ccccc} y(t) & y_{1}(t) & y_{2}(t) & \cdots & y_{n}(t) \\ y^{\prime}(t) & y_{1}^{\prime}(t) & y_{2}^{\prime}(t) & \cdots & y_{n}^{\prime}(t) \\ \vdots & \vdots & \vdots & & \vdots \\ y^{(n)}(t) & y_{1}^{(n)}(t) & y_{2}^{(n)}(t) & \cdots & y_{n}^{(n)}(t) \end{array}\right) . $$ The preceding determinant is called the Wronskian of $y, y_{1}, \ldots, y_{n}$. (a) Prove that $\mathrm{T}: \mathrm{C}^{\infty} \rightarrow \mathrm{C}^{\infty}$ is a linear transformation. (b) Prove that \(\mathbf{N}(\mathbf{T})\) contains span $\left(\left\\{y_{1}, y_{2}, \ldots, y_{n}\right\\}\right)$.

Find the dimension and a basis of the subspace \(W\) of \(\mathbf{P}_{3}(t)\) spanned by \\[u=t^{3}+2 t^{2}-3 t+4, \quad v=2 t^{3}+5 t^{2}-4 t+7, \quad w=t^{3}+4 t^{2}+t+2\\]

Let \(V\) be the set of infinite sequences \(\left(a_{1}, a_{2}, \ldots\right)\) in a field \(K .\) Show that \(V\) is a vector space over \(K\) with addition and scalar multiplication defined by \\[\left(a_{1}, a_{2}, \ldots\right)+\left(b_{1}, b_{2}, \ldots\right)=\left(a_{1}+b_{1}, a_{2}+b_{2}, \ldots\right) \quad \text { and } \quad k\left(a_{1}, a_{2}, \ldots\right)=\left(k a_{1}, k a_{2}, \ldots\right)\\]

Evaluate the determinant of the following matrices in the manner indicated. (a) $\left(\begin{array}{rrr}0 & 1 & 2 \\ -1 & 0 & -3 \\ 2 & 3 & 0\end{array}\right)$ along the first row (b) $\left(\begin{array}{rrr}1 & 0 & 2 \\ 0 & 1 & 5 \\ -1 & 3 & 0\end{array}\right)$ along the first column (c) $\left(\begin{array}{rrr}0 & 1 & 2 \\ -1 & 0 & -3 \\ 2 & 3 & 0\end{array}\right)$ along the second column (d) $\left(\begin{array}{rrr}1 & 0 & 2 \\ 0 & 1 & 5 \\ -1 & 3 & 0\end{array}\right)$ along the third row (e) $\left(\begin{array}{ccc}0 & 1+i & 2 \\ -2 i & 0 & 1-i \\ 3 & 4 i & 0\end{array}\right)$ along the third column (f) $\quad\left(\begin{array}{ccc}i & 2+i & 0 \\ -1 & 3 & 2 i \\ 0 & -1 & 1-i\end{array}\right)$ along the third row (g) $\left(\begin{array}{rrrr}0 & 2 & 1 & 3 \\ 1 & 0 & -2 & 2 \\ 3 & -1 & 0 & 1 \\ -1 & 1 & 2 & 0\end{array}\right)$ along the fourth column (h) $\left(\begin{array}{rrrr}1 & -1 & 2 & -1 \\ -3 & 4 & 1 & -1 \\ 2 & -5 & -3 & 8 \\ -2 & 6 & -4 & 1\end{array}\right)$ along the fourth row
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