Category Archives: Chapter 3

Chapter 3 Exercise F

1. Solution: For any $\vp\in\ca L(V,\mb F)$, if $\dim \m{range} \vp=0$, then $\vp$ is the zero map. If $\dim \m{range} \vp=1$, then $\vp$ is surjective since $\dim\mb F=1$. Moreover, $\dim \m{range} \vp\leqslant \dim \mb F=1$. Hence, that is all the … Continue reading

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Chapter 3 Exercise E

Exercises 1,2 and 4. For Problem 2, please also see Carson Rogers’s comment. 4. Solution: For any $f\in \ca L(V_1\times \cdots\times V_m,W)$ and given $i\in \{1,\cdots,m\}$, define $f_i:V_i\to W$ by the rule:\[f_i(v_i)=f(0,\cdots,0,v_i,0,\cdots,0),\]where $v_i$ sits in the $i$-th slot. One can … Continue reading

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Chapter 3 Exercise D

1. Suppose $T\in\ca L(U, V)$ and $S\in\ca L(V, W)$ are both invertible linear maps. Prove that $ST\in\ca L(U, W)$ / is invertible and that $(ST)^{-1}=T^{-1}S^{-1}$. Solution: See Linear Algebra Done Right Solution Manual Chapter 3 Problem 22. It is almost … Continue reading

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Chapter 3 Exercise C

1. Suppose $V$ and $W$ are finite-dimensional and $T\in\ca L(V,W)$. Show that with respect to each choice of bases of $V$ and $W$, the matrix of $T$ has at least $\dim \m{range} T$ nonzero entries. Solution: Suppose for some basis … Continue reading

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Chapter 3 Exercise B

1. Give an example of a linear map $T$ such that $\dim \mathrm{null} T=3$ and $\dim \mathrm{range} T = 2$. Solution: Assume $V$ is 5-dimensional vector space with a basis $e_1$, $\cdots$, $e_5$. Define $T\in\ca L(V,V)$ by \[Te_1=e_1,Te_2=e_2,Te_3=Te_4=Te_5=0.\]Then $\mathrm{null} T=\mathrm{span}(e_3,e_4,e_5)$, … Continue reading

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Chapter 3 Exercise A

1. Suppose $b,c\in \R$. Define $T: \R^3 \to \R^2$ by \[T(x, y, z)= (2x-4y +3z + b,6x +cxyz).\] Show that $T$ is linear if and only if $b=c=0$. Solution: If $T$ is linear, then \[(0,0)=T(0,0,0)=(b,0)\]by 3.11, hence $b=0$. We also … Continue reading

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