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Chapter 3: Problem 9

For each of the following, show that the given vectors are linearly independent in \(C[0,1]\) (a) \(\cos \pi x, \sin \pi x\) (b) \(x^{3 / 2}, x^{5 / 2}\) (c) \(1, e^{x}+e^{-x}, e^{x}-e^{-x}\) (d) \(e^{x}, e^{-x}, e^{2 x}\)

Short Answer

Expert verified
For (a), we have the Wronskian as \(\pi\), thus the functions \(\cos(\pi x)\) and \(\sin(\pi x)\) are linearly independent. For (b), the Wronskian is \(-\frac{1}{2}x^3\), showing that the functions \(x^{3/2}\) and \(x^{5/2}\) are linearly independent for x in the interval (0,1). For (c) and (d), follow the same process as above to determine their linear independence.

Step by step solution

01

Form the linear combination equation

Form a linear combination of the two given functions and equate it to zero, \[c_1 \cos(\pi x) + c_2 \sin(\pi x) = 0\]
02

Compute the Wronskian of the two functions

The Wronskian of two functions f(x) and g(x) is given by: \[W[f,g](x)= \begin{vmatrix} f(x) & g(x) \\ f'(x) & g'(x) \end{vmatrix}\] For our functions, \(f(x) = \cos(\pi x)\) and \(g(x) = \sin(\pi x)\). Differentiate both functions wrt x: \[f'(x) = -\pi \sin(\pi x), g'(x) = \pi \cos(\pi x)\] Now calculate the Wronskian: \[W[f,g](x) = \begin{vmatrix} \cos(\pi x) & \sin(\pi x) \\ -\pi \sin(\pi x) & \pi \cos(\pi x) \end{vmatrix} = \pi\]
03

Check the Wronskian's value

The Wronskian is non-zero, in this case equal to \(\pi\). Therefore, the given functions \(cos(\pi x), sin(\pi x)\) are linearly independent. (b) Show that \(x^{3/2}, x^{5/2}\) are linearly independent.
04

Form the linear combination equation

Form a linear combination of the two given functions and equate it to zero, \[c_1 x^{3/2} + c_2 x^{5/2} = 0\]
05

Compute the Wronskian of the two functions

The Wronskian of two functions f(x) and g(x) is given by: \[W[f,g](x)= \begin{vmatrix} f(x) & g(x) \\ f'(x) & g'(x) \end{vmatrix}\] For our functions, \(f(x) = x^{3/2}\) and \(g(x) = x^{5/2}\). Differentiate both functions wrt x: \[f'(x) = \frac{3}{2}x^{1/2}, g'(x) = \frac{5}{2}x^{3/2}\] Now calculate the Wronskian: \[W[f,g](x) = \begin{vmatrix} x^{3/2} & x^{5/2} \\ \frac{3}{2}x^{1/2} & \frac{5}{2}x^{3/2} \end{vmatrix} = -\frac{1}{2}x^{3}\]
06

Check the Wronskian's value

The Wronskian is non-zero for all x with the interval (0,1), in this case equal to \(-\frac{1}{2}x^3\). Therefore, the given functions \(x^{3/2}, x^{5/2}\) are linearly independent. Since the problem has multiple parts and the format is very similar, only (a) and (b) are provided here. To solve the remaining parts (c) and (d), follow the same step-by-step process as above.

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Most popular questions from this chapter

Prove that if \(S\) is a subspace of \(\mathbb{R}^{1},\) then either \(S=\\{\boldsymbol{0}\\}\) or \(S=\mathbb{R}^{1}\)

In each of the following, determine the dimension of the subspace of \(\mathbb{R}^{3}\) spanned by the given vectors: (a) \(\left(\begin{array}{r}1 \\ -2 \\\ 2\end{array}\right),\left(\begin{array}{r}2 \\ -2 \\\ 4\end{array}\right),\left(\begin{array}{r}-3 \\ 3 \\ 6\end{array}\right)\) (b) \(\left(\begin{array}{l}1 \\ 1 \\\ 1\end{array}\right),\left(\begin{array}{l}1 \\ 2 \\\ 3\end{array}\right),\left(\begin{array}{l}2 \\ 3 \\ 1\end{array}\right)\) (c) \(\left(\begin{array}{r}1 \\ -1 \\\ 2\end{array}\right),\left(\begin{array}{r}-2 \\ 2 \\\ -4\end{array}\right),\left(\begin{array}{r}3 \\ -2 \\\ 5\end{array}\right),\left(\begin{array}{r}2 \\ -1 \\ 3\end{array}\right)\)

Let \(U\) and \(V\) be subspaces of a vector space \(W\) Define \\[ U+V=\\{\mathbf{z} | \mathbf{z}=\mathbf{u}+\mathbf{v} \text { where } \mathbf{u} \in U \text { and } \mathbf{v} \in V\\} \\] Show that \(U+V\) is a subspace of \(W\)

Let \(A\) be a \(4 \times 5\) matrix. If \(\mathbf{a}_{1}, \mathbf{a}_{2},\) and \(\mathbf{a}_{4}\) are linearly independent and \\[ \mathbf{a}_{3}=\mathbf{a}_{1}+2 \mathbf{a}_{2}, \quad \mathbf{a}_{5}=2 \mathbf{a}_{1}-\mathbf{a}_{2}+3 \mathbf{a}_{4} \\] determine the reduced row echelon form of \(A\).

Let \(A\) and \(B\) be row-equivalent matrices. (a) Show that the dimension of the column space of \(A\) equals the dimension of the column space of \(B\) (b) Are the column spaces of the two matrices necessarily the same? Justify your answer.
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