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Chapter 5: Problem 11

Show that the following mappings are not linear: (a) \(F: \mathbf{R}^{2} \rightarrow \mathbf{R}^{2}\) defined by \(F(x, y)=(x y, x)\) (b) \(F: \mathbf{R}^{2} \rightarrow \mathbf{R}^{3}\) defined by \(F(x, y)=(x+3,2 y, x+y)\) (c) \(F: \mathbf{R}^{3} \rightarrow \mathbf{R}^{2}\) defined by \(F(x, y, z)=(|x|, y+z)\)

Short Answer

Expert verified
In summary, all three mappings are not linear transformations as they fail to satisfy the properties of addition and/or scalar multiplication. Specifically: (a) Fails the addition property: (x1 + x2)(y1 + y2) 鈮� x1 * y1 + x2 * y2 (b) Fails both addition and scalar multiplication properties: x1 + x2 + 3 鈮� x1 + x2 + 6, and c * x + 3 鈮� c * x + 3 * c (c) Fails both addition and scalar multiplication properties: (|x1| + |x2|) 鈮� |(x1 + x2)|, and c * |x| 鈮� |c * x|

Step by step solution

01

(a) Testing Mapping F for addition and scalar multiplication

: We start by testing the addition property for mapping (a): F(x1 + x2, y1 + y2) = ((x1 + x2)(y1 + y2), (x1 + x2)) To show it is not a linear mapping, we put F(u) + F(v): F(x1, y1) + F(x2, y2) = ((x1 * y1, x1) + (x2 * y2, x2)) = ((x1 * y1 + x2 * y2), (x1 + x2)) If they don't match, we can conclude the mapping is not linear in addition: (x1 + x2)(y1 + y2) 鈮� x1 * y1 + x2 * y2 Now, we test for scalar multiplication: F(c * x, c * y) = (c * x * c * y, c * x) To show it is not a linear mapping, we put c * F(u): c * F(x, y) = c * (x * y, x) = (c * x * y, c * x) Comparing these, we see that they match: c * x * c * y = c * x * y As one of the properties (addition) is not satisfied, we can conclude that mapping (a) is not linear.
02

(b) Testing Mapping F for addition and scalar multiplication

: We start by testing the addition property for mapping (b): F(x1 + x2, y1 + y2) = (x1 + x2 + 3, 2 * (y1 + y2), (x1 + x2) + (y1 + y2)) To show it is linear, we put F(u) + F(v): F(x1, y1) + F(x2, y2) = ((x1 + 3, 2 * y1, x1 + y1) + (x2 + 3, 2 * y2, x2 + y2)) = ((x1 + x2 + 6), (2 * (y1 + y2), (x1 + x2) + (y1 + y2)) Comparing these, we see that they don't match: x1 + x2 + 3 鈮� x1 + x2 + 6 Now, we test for scalar multiplication: F(c * x, c * y) = (c * x + 3, 2 * c * y, c * x + c * y) To show it is linear, we put c * F(u): c * F(x, y) = c * (x + 3, 2 * y, x + y) = (c * x + 3 * c, 2 * c * y, c * (x + y)) Comparing these, we see that they don't match: c * x + 3 鈮� c * x + 3 * c As both properties (addition and scalar multiplication) are not satisfied, we can conclude that mapping (b) is not linear.
03

(c) Testing Mapping F for addition and scalar multiplication

: We start by testing the addition property for mapping (c): F(x1 + x2, y1 + y2, z1 + z2) = (|(x1 + x2)|, (y1 + y2) + (z1 + z2)) To show it is linear, we put F(u) + F(v): F(x1, y1, z1) + F(x2, y2, z2) = (|x1|, y1 + z1) + (|x2|, y2 + z2) = (|x1| + |x2|, (y1 + z1) + (y2 + z2)) Comparing these, we see that they don't match: (|x1| + |x2|) 鈮� |(x1 + x2)| Now, we test for scalar multiplication: F(c * x, c * y, c * z) = (|c * x|, c * y + c * z) To show it is linear, we put c * F(u): c * F(x, y, z) = c * (|x|, y + z) = (c * |x|, c * (y + z)) Comparing these, we see that they don't match: c * |x| 鈮� |c * x| As both properties (addition and scalar multiplication) are not satisfied, we can conclude that mapping (c) is not linear.

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Most popular questions from this chapter

Let \(A\) be an \(n \times n\) matrix. Prove that $$ \operatorname{dim}\left(\operatorname{span}\left(\left\\{I_{n}, A, A^{2}, \ldots\right\\}\right)\right) \leq n . $$

For each of the following linear operators \(T\) on a vector space \(V\) and ordered bases \(\beta\), compute \([T]_{\beta}\), and determine whether \(\beta\) is a basis consisting of eigenvectors of \(T\). (a) $\mathrm{V}=\mathrm{R}^{2}, \mathrm{~T}\left(\begin{array}{l}a \\\ b\end{array}\right)=\left(\begin{array}{l}10 a-6 b \\ 17 a-10 b\end{array}\right)\(, and \)\beta=\left\\{\left(\begin{array}{l}1 \\\ 2\end{array}\right),\left(\begin{array}{l}2 \\ 3\end{array}\right)\right\\}$ (b) $\quad \mathrm{V}=\mathrm{P}_{1}(R), \mathrm{T}(a+b x)=(6 a-6 b)+(12 a-11 b) x$, and $$ \beta=\\{3+4 x, 2+3 x\\} $$ (c) $\mathrm{V}=\mathrm{R}^{3}, \mathrm{~T}\left(\begin{array}{l}a \\ b \\\ c\end{array}\right)=\left(\begin{array}{r}3 a+2 b-2 c \\ -4 a-3 b+2 c \\\ -c\end{array}\right)$, and $$ \beta=\left\\{\left(\begin{array}{l} 0 \\ 1 \\ 1 \end{array}\right),\left(\begin{array}{r} 1 \\ -1 \\ 0 \end{array}\right),\left(\begin{array}{l} 1 \\ 0 \\ 2 \end{array}\right)\right\\} $$ (d) \(\mathrm{V}=\mathrm{P}_{2}(R), \mathrm{T}\left(a+b x+c x^{2}\right)=\) $$ (-4 a+2 b-2 c)-(7 a+3 b+7 c) x+(7 a+b+5 c) x^{2}, $$ and \(\beta=\left\\{x-x^{2},-1+x^{2},-1-x+x^{2}\right\\}\) (e) $\mathrm{V}=\mathrm{P}_{3}(R), \mathrm{T}\left(a+b x+c x^{2}+d x^{3}\right)=$ $$ -d+(-c+d) x+(a+b-2 c) x^{2}+(-b+c-2 d) x^{3}, $$ and \(\beta=\left\\{1-x+x^{3}, 1+x^{2}, 1, x+x^{2}\right\\}\) (f) $\mathrm{V}=\mathrm{M}_{2 \times 2}(R), \mathrm{T}\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)=\left(\begin{array}{ll}-7 a-4 b+4 c-4 d & b \\\ -8 a-4 b+5 c-4 d & d\end{array}\right)$, and $$ \beta=\left\\{\left(\begin{array}{ll} 1 & 0 \\ 1 & 0 \end{array}\right),\left(\begin{array}{rr} -1 & 2 \\ 0 & 0 \end{array}\right),\left(\begin{array}{ll} 1 & 0 \\ 2 & 0 \end{array}\right),\left(\begin{array}{rr} -1 & 0 \\ 0 & 2 \end{array}\right)\right\\} $$

In 1975 , the automobile industry determined that \(40 \%\) of American car owners drove large cars, \(20 \%\) drove intermediate-sized cars, and \(40 \%\) drove small cars. A second survey in 1985 showed that \(70 \%\) of the largecar owners in 1975 still owned large cars in 1985 , but \(30 \%\) had changed to an intermediate-sized car. Of those who owned intermediate-sized cars in $1975,10 \%\( had switched to large cars, \)70 \%$ continued to drive intermediate-sized cars, and \(20 \%\) had changed to small cars in 1985 . Finally, of the small- car owners in \(1975,10 \%\) owned intermediate-sized cars and \(90 \%\) owned small cars in 1985 . Assuming that these trends continue, determine the percentages of Americans who own cars of each size in 1995 and the corresponding eventual percentages.

Find a \(2 \times 2\) matrix \(A\) that maps (a) \(\quad(1,3)^{T}\) and \((1,4)^{T}\) into \((-2,5)^{T}\) and \((3,-1)^{T},\) respectively (b) \((2,-4)^{T}\) and \((-1,2)^{T}\) into \((1,1)^{T}\) and \((1,3)^{T},\) respectively.

Find \(F(a, b),\) where the linear map \(F: \mathbf{R}^{2} \rightarrow \mathbf{R}^{2}\) is defined by \(F(1,2)=(3,-1)\) and \(F(0,1)=(2,1).\)
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