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Chapter 1: Problem 15

Is the set of all differentiable real-valued functions defined on \(R\) a subspace of \(C(R)\) ? Justify your answer.

Short Answer

Expert verified
Yes, the set of all differentiable real-valued functions defined on \(R\) is a subspace of \(C(R)\) because it contains the zero function, and it is closed under both addition and scalar multiplication.

Step by step solution

01

Check if the zero function is in the set of differentiable functions

The zero function is defined as \(f(x) = 0, \forall x \in R\). The derivative of the zero function with respect to \(x\) is \(f'(x) = 0, \forall x \in R\), which means the zero function is differentiable. Thus, the zero function is in the set of differentiable real-valued functions.
02

Check if the set is closed under addition

Let \(f(x)\) and \(g(x)\) be any two differentiable real-valued functions defined on \(R\). We need to verify if their sum, \(f(x) + g(x)\), is also differentiable. The sum rule of differentiation states that \((f(x) + g(x))' = f'(x) + g'(x)\), which means that the derivative exists and is continuous. Since \(f(x) + g(x)\) is differentiable, the set is closed under addition.
03

Check if the set is closed under scalar multiplication

Let \(f(x)\) be a differentiable real-valued function defined on \(R\) and let \(c\) be any scalar. We need to verify if the scalar multiplication of \(f(x)\) and \(c\), which is \(cf(x)\), is differentiable. The scalar multiplication rule of differentiation states that \((cf(x))' = cf'(x)\). Since the derivative exists and is continuous, the function \(cf(x)\) is differentiable. Therefore, the set is closed under scalar multiplication. Since the set of differentiable real-valued functions defined on \(R\) satisfies the subspace criteria, it is a subspace of \(C(R)\).

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Most popular questions from this chapter

Find a unit vector \(u\) orthogonal to: (a) \(v=[1,2,3]\) and \(w=[1,-1,2]\) (b) \(v=3 \mathbf{i}-\mathbf{j}+2 \mathbf{k}\) and \(w=4 \mathbf{i}-2 \mathbf{j}-\mathbf{k}\)

Show that if \(S_{1}\) and \(S_{2}\) are subsets of a vector space \(\mathrm{V}\) such that \(S_{1} \subseteq S_{2}\), then $\operatorname{span}\left(S_{1}\right) \subseteq \operatorname{span}\left(S_{2}\right) .\( In particular, if \)S_{1} \subseteq S_{2}\( and \)\operatorname{span}\left(S_{1}\right)=\mathrm{V}$, deduce that span \(\left(S_{2}\right)=\mathrm{V}\). Visit goo.gl/Fi8Epr for a solution.

A real-valued function \(f\) defined on the real line is called an even function if \(f(-t)=f(t)\) for each real number \(t\). Prove that the set of even functions defined on the real line with the operations of addition and scalar multiplication defined in Example 3 is a vector space.

Label the following statements as true or false. (a) If \(V\) is a vector space and \(W\) is a subset of \(V\) that is a vector space, then \(W\) is a subspace of \(V\). (b) The empty set is a subspace of every vector space. (c) If \(V\) is a vector space other than the zero vector space, then \(V\) contains a subspace \(W\) such that \(W \neq V\). (d) The intersection of any two subsets of \(V\) is a subspace of \(V\). (e) An \(n \times n\) diagonal matrix can never have more than \(n\) nonzero entries. (f) The trace of a square matrix is the product of its diagonal entries. (g) Let \(\mathrm{W}\) be the \(x y\)-plane in \(\mathrm{R}^{3}\); that is, $\mathrm{W}=\left\\{\left(a_{1}, a_{2}, 0\right): a_{1}, a_{2} \in R\right\\}\(. Then \)W=R^{2}$.

Determine whether the following sets are linearly dependent or linearly independent. (a) $\left\\{\left(\begin{array}{rr}1 & -3 \\ -2 & 4\end{array}\right),\left(\begin{array}{rr}-2 & 6 \\ 4 & -8\end{array}\right)\right\\}\( in \)\mathrm{M}_{2 \times 2}(R)$ (b) $\left\\{\left(\begin{array}{rr}1 & -2 \\ -1 & 4\end{array}\right),\left(\begin{array}{rr}-1 & 1 \\ 2 & -4\end{array}\right)\right\\}\( in \)\mathrm{M}_{2 \times 2}(R)$ (c) \(\left\\{x^{3}+2 x^{2},-x^{2}+3 x+1, x^{3}-x^{2}+2 x-1\right\\}\) in \(\mathrm{P}_{3}(R)\) (d) \(\left\\{x^{3}-x, 2 x^{2}+4,-2 x^{3}+3 x^{2}+2 x+6\right\\}\) in \(\mathrm{P}_{3}(R)\) (e) \(\\{(1,-1,2),(1,-2,1),(1,1,4)\\}\) in \(\mathbf{R}^{3}\) (f) \(\\{(1,-1,2),(2,0,1),(-1,2,-1)\\}\) in \(\mathbf{R}^{3}\) (g) $\left\\{\left(\begin{array}{rr}1 & 0 \\ -2 & 1\end{array}\right),\left(\begin{array}{rr}0 & -1 \\ 1 & 1\end{array}\right),\left(\begin{array}{rr}-1 & 2 \\ 1 & 0\end{array}\right),\left(\begin{array}{rr}2 & 1 \\ -4 & 4\end{array}\right)\right\\}\( in \)\mathrm{M}_{2 \times 2}(R)$ (h) $\left\\{\left(\begin{array}{rr}1 & 0 \\ -2 & 1\end{array}\right),\left(\begin{array}{rr}0 & -1 \\ 1 & 1\end{array}\right),\left(\begin{array}{rr}-1 & 2 \\ 1 & 0\end{array}\right),\left(\begin{array}{rr}2 & 1 \\ 2 & -2\end{array}\right)\right\\}\( in \)\mathrm{M}_{2 \times 2}(R)$ (i) \(\left\\{x^{4}-x^{3}+5 x^{2}-8 x+6,-x^{4}+x^{3}-5 x^{2}+5 x-3\right.\), \(\left.x^{4}+3 x^{2}-3 x+5,2 x^{4}+3 x^{3}+4 x^{2}-x+1, x^{3}-x+2\right\\}\) in \(\mathrm{P}_{4}(R)\) (j) \(\left\\{x^{4}-x^{3}+5 x^{2}-8 x+6,-x^{4}+x^{3}-5 x^{2}+5 x-3,\right.\), \(\left.x^{4}+3 x^{2}-3 x+5,2 x^{4}+x^{3}+4 x^{2}+8 x\right\\}\) in \(\mathrm{P}_{4}(R)\) \({ }^{3}\) The computations in Exercise \(2(\mathrm{~g}),(\mathrm{h}),(\mathrm{i})\), and \((\mathrm{j})\) are tedious unless technology is used.
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