/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 9 In this exercise, \(T: R^{2} \ri... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 9

In this exercise, \(T: R^{2} \rightarrow R^{2}\) is a function. For each of the following parts, state why \(\mathrm{T}\) is not linear. (a) \(\mathrm{T}\left(a_{1}, a_{2}\right)=\left(1, a_{2}\right)\) (b) \(\mathrm{T}\left(a_{1}, a_{2}\right)=\left(a_{1}, a_{1}^{2}\right)\) (c) \(\mathrm{T}\left(a_{1}, a_{2}\right)=\left(\sin a_{1}, 0\right)\) (d) $\mathrm{T}\left(a_{1}, a_{2}\right)=\left(\left|a_{1}\right|, a_{2}\right)$ (e) \(\mathrm{T}\left(a_{1}, a_{2}\right)=\left(a_{1}+1, a_{2}\right)\)

Short Answer

Expert verified
In summary, the given transformations T are not linear because they do not satisfy either the Additivity or Homogeneity property, as shown in each part: (a) Additivity is not satisfied. (b) Homogeneity is not satisfied. (c) Homogeneity is not satisfied. (d) Homogeneity is not satisfied for some negative scalar values. (e) Additivity is not satisfied.

Step by step solution

01

Part (a)

Let \(T(a_1, a_2)=(1, a_2)\). To check Additivity, let's take two vectors, \(u=(a_1, a_2)\) and \(v=(b_1, b_2)\), and then calculate \(T(u+v)\) and \(T(u)+T(v)\): \(T(u+v)=T(a_1+b_1, a_2+b_2)=(1, a_2+b_2)\) \(T(u)+T(v)=(1, a_2)+(1, b_2)=(2, a_2+b_2)\) Since \(T(u+v) \neq T(u)+T(v)\), Additivity is not satisfied. Therefore, T is not linear in this case.
02

Part (b)

Let \(T(a_1, a_2)=(a_1, a_1^2)\). To check Homogeneity, let's take a vector, \(v=(a_1, a_2)\), and a scalar, c, and then calculate \(T(cv)\) and \(cT(v)\): \(T(cv)=T(ca_1, ca_2)=(ca_1, (ca_1)^2)\) \(cT(v)=c(a_1, a_1^2)=(ca_1, ca_1^2)\) Since \(T(cv) \neq cT(v)\), Homogeneity is not satisfied. Therefore, T is not linear in this case.
03

Part (c)

Let \(T(a_1, a_2)=(\sin a_1, 0)\). To check Homogeneity, let's take a vector, \(v=(a_1, a_2)\), and a scalar, c, and then calculate \(T(cv)\) and \(cT(v)\): \(T(cv)=T(ca_1, ca_2)=(\sin (ca_1), 0)\) \(cT(v)=c(\sin a_1, 0)=(c\sin a_1, 0)\) Since \(T(cv) \neq cT(v)\), Homogeneity is not satisfied. Therefore, T is not linear in this case.
04

Part (d)

Let \(T(a_1, a_2)=(|a_1|, a_2)\). To check Homogeneity, let's take a vector, \(v=(a_1, a_2)\), and a scalar, c, and then calculate \(T(cv)\) and \(cT(v)\): \(T(cv)=T(ca_1, ca_2)=(|ca_1|, ca_2)\) \(cT(v)=c(|a_1|, a_2)=(c|a_1|, ca_2)\) Since \(T(cv) \neq cT(v)\) for some negative values of c, Homogeneity is not satisfied. Therefore, T is not linear in this case.
05

Part (e)

Let \(T(a_1, a_2)=(a_1+1, a_2)\). To check Additivity, let's take two vectors, \(u=(a_1, a_2)\) and \(v=(b_1, b_2)\), and then calculate \(T(u+v)\) and \(T(u)+T(v)\): \(T(u+v)=T(a_1+b_1, a_2+b_2)=(a_1+b_1+1, a_2+b_2)\) \(T(u)+T(v)=(a_1+1, a_2)+(b_1+1, b_2)=(a_1+b_1+2, a_2+b_2)\) Since \(T(u+v) \neq T(u)+T(v)\), Additivity is not satisfied. Therefore, T is not linear in this case.

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Most popular questions from this chapter

Let \(f(x)\) and \(g(x)\) be polynomials and let \(A\) be a square matrix. Prove (a) \((f+g)(A)=f(A)+g(A)\) (b) \((f \cdot g)(A)=f(A) g(A)\) (c) \(f(A) g(A)=g(A) f(A)\)

Let \(V\) be a nonzero vector space, and let \(W\) be a proper subspace of \(V\) (i.e., \(W \neq V\) ). (a) Let \(\mathrm{g} \in \mathrm{W}^{*}\) and \(v \in \mathrm{V}\) with $v \notin \mathrm{W}\(. Prove that for any scalar \)a\( there exists a function \)\mathrm{f} \in \mathrm{V}^{*}\( such that \)\mathrm{f}(v)=a$ and \(\mathrm{f}(x)=\mathrm{g}(x)\) for all \(x\) in W. Hint: For the infinite- dimensional case, use Exercise 4 of Section \(1.7\) and Exercise 35 of Section 2.1. (b) Use (a) to prove there exists a nonzero linear functional $f \in \mathrm{V}^{*}\( such that \)\mathrm{f}(x)=0\( for all \)x \in \mathrm{W}$.

Let \(A\) and \(B\) be \(n \times n\) matrices such that \(A B\) is invertible. (a) Prove that \(A\) and \(B\) are invertible. Hint: See Exercise 12 of Section 2.3. (b) Give an example to show that a product of nonsquare matrices can be invertible even though the factors, by definition, are not.

Repeat Example 7 with the polynomial \(p(x)=1+x+2 x^{2}+x^{3}\).

A differential equation $$ y^{(n)}+a_{n-1} y^{(n-1)}+\cdots+a_{1} y^{(1)}+a_{0} y=x $$ is called a nonhomogeneous linear differential equation with constant coefficients if the \(a_{i}\) 's are constant and \(x\) is a function that is not identically zero.(a) Prove that for any \(x \in{C}^{\infty}\) there exists $y \in{C}^{\infty}\( such that \)y$ is a solution to the differential equation. Hint: Use Lemma 1 to Theorem \(2.32\) to show that for any polynomial \(p(t)\), the linear operator $p(\mathrm{D}): \mathrm{C}^{\infty} \rightarrow \mathrm{C}^{\infty}$ is onto. (b) Let \(V\) be the solution space for the homogeneous linear equation $$ y^{(n)}+a_{n-1} y^{(n-1)}+\cdots+a_{1} y^{(1)}+a_{0} y=0 . $$ Prove that if \(z\) is any solution to the associated nonhomogeneous linear differential equation, then the set of all solutions to the nonhomogeneous linear differential equation is $$ \\{z+y: y \in \mathrm{V}\\} . $$
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