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Chapter 2: Problem 18
Repeat Example 7 with the polynomial \(p(x)=1+x+2 x^{2}+x^{3}\).
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For each of the following parts, determine whether the statement is true or false. Justify your claim with either a proof or a counterexample, whichever is appropriate. (a) Any finite-dimensional subspace of \(C^{\infty}\) is the solution space of a homogeneous linear differential equation with constant coefficients. (b) There exists a homogeneous linear differential equation with constant coefficients whose solution space has the basis \(\left\\{t, t^{2}\right\\}\). (c) For any homogeneous linear differential equation with constant coefficients, if \(x\) is a solution to the equation, so is its derivative \(x^{\prime}\). Given two polynomials \(p(t)\) and \(q(t)\) in \(\mathrm{P}(C)\), if $x \in \mathrm{N}(p(\mathrm{D}))\( and \)y \in\( \)\mathrm{N}(q(\mathrm{D}))$, then (d) \(x+y \in \mathrm{N}(p(\mathrm{D}) q(\mathrm{D}))\). (e) \(x y \in \mathrm{N}(p(\mathrm{D}) q(\mathrm{D}))\).
Let \(U=\left[\begin{array}{ccccc}1 & 2 & 1 & 0 & 0 & 0 \\ \frac{3}{0} & \frac{4}{1} & \frac{1}{1} & \frac{0}{5} & -\frac{0}{1} & -\frac{0}{2} \\ 0 & 0 & 1 & 1 & 2 \\ 0 & 0 & 3 & 4 & 1\end{array}\right]\) and \(V=\left[\begin{array}{ccccc}3 & -2 & 1 & 0 & 0 \\ \frac{2}{0} & -\frac{4}{0} & -\frac{1}{1} & \frac{0}{1} & -\frac{0}{2} \\ 0 & 0 & 1 & 2 & -3 \\ 0 & 0 & 1-4 & 1\end{array}\right]\) (a) Find \(U V\) using block multiplication. (b) Are \(U\) and \(V\) block diagonal matrices? (c) Is \(U V\) block diagonal?
Let \(A=\left[\begin{array}{ll}1 & 2 \\ 3 & 6\end{array}\right] .\) Find a \(2 \times 3\) matrix \(B\) with distinct nonzero entries such that \(A B=0.\)
Let \(\mathrm{V}=\mathrm{P}_{n}(F)\), and let \(c_{0}, c_{1}, \ldots, c_{n}\) be distinct scalars in \(F\). (a) For \(0 \leq i \leq n\), define \(\mathrm{f}_{i} \in \mathrm{V}^{*}\) by \(\mathrm{f}_{i}(p(x))=p\left(c_{i}\right)\). Prove that \(\left\\{\mathrm{f}_{0}, \mathrm{f}_{1}, \ldots, \mathrm{f}_{n}\right\\}\) is a basis for \(\mathrm{V}^{*}\). Hint: Apply any linear combination of this set that equals the zero transformation to \(p(x)=\) \(\left(x-c_{1}\right)\left(x-c_{2}\right) \cdots\left(x-c_{n}\right)\), and deduce that the first coefficient is zero. (b) Use the corollary to Theorem \(2.26\) and (a) to show that there exist unique polynomials \(p_{0}(x), p_{1}(x), \ldots, p_{n}(x)\) such that \(p_{i}\left(c_{j}\right)=\delta_{i j}\) for \(0 \leq i \leq n\). These polynomials are the Lagrange polynomials defined in Section 1.6. (c) For any scalars \(a_{0}, a_{1}, \ldots, a_{n}\) (not necessarily distinct), deduce that there exists a unique polynomial \(q(x)\) of degree at most \(n\) such that \(q\left(c_{i}\right)=a_{i}\) for \(0 \leq i \leq n .\) In fact, $$ q(x)=\sum_{i=0}^{n} a_{i} p_{i}(x) $$ (d) Deduce the Lagrange interpolation formula: $$ p(x)=\sum_{i=0}^{n} p\left(c_{i}\right) p_{i}(x) $$ for any \(p(x) \in \mathrm{V}\). (e) Prove that $$ \int_{a}^{b} p(t) d t=\sum_{i=0}^{n} p\left(c_{i}\right) d_{i} $$ where $$ d_{i}=\int_{a}^{b} p_{i}(t) d t $$ Suppose now that $$ c_{i}=a+\frac{i(b-a)}{n} \text { for } i=0,1, \ldots, n \text {. } $$ For \(n=1\), the preceding result yields the trapezoidal rule for evaluating the definite integral of a polynomial. For \(n=2\), this result yields Simpson's rule for evaluating the definite integral of a polynomial.
Let \(A\) be an \(n \times n\) matrix. Prove that \(A\) is a diagonal matrix if and only if \(A_{i j}=\delta_{i j} A_{i j}\) for all \(i\) and \(j\).
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