Chapter 1Exercises 1C13Prove that the union of three subspaces ofVVVis a subspace ofVVVif and only if one of the subspaces contains the other two.Proof.(⇒\Rightarrow⇒) Let the three subspaces beA,B,CA,B,CA,B,C. There are two cases:A⊆BA\subseteq BA⊆B. By the result of Exercise 12, eitherB⊆CB\subseteq CB⊆CorC⊆BC\subseteq BC⊆B, in both cases we are done.A⊈B∧B⊈AA\not\subseteq B\land B\not\subseteq AA⊆B∧B⊆A. LetAL(a1,…,ai,d1,…,dk),BL(b1,…,bj,d1,…,dk)AL(a_1,\ldots,a_i,d_1,\ldots,d_k),BL(b_1,\ldots,b_j,d_1,\ldots,d_k)AL(a1,…,ai,d1,…,dk),BL(b1,…,bj,d1,…,dk)wherei0,j0,k≥0i0,j0,k\ge0i0,j0,k≥0. Soa1∈A−B,b1∈B−Aa_1\in A-B,b_1\in B-Aa1∈A−B,b1∈B−A. The smallest subspace containing bothAAAandBBBisABABAB. By the result of Exercise 12, eitherAB⊆CAB\subseteq CAB⊆C(done) orC⊊ABC\subsetneq ABC⊊AB. In the latter case, the question is whetherC⊇SC\supseteq SC⊇S, whereSAB−(A∪B)L(a1,…,ai,b1,…,bj,d1,…,dk)−L(a1,…,ai,d1,…,dk)−L(b1,…,bj,d1,…,dk)?\begin{align*}SAB-(A\cup B)\\ L(a_1,\ldots,a_i,b_1,\ldots,b_j,d_1,\ldots,d_k)\\ \qquad-L(a_1,\ldots,a_i,d_1,\ldots,d_k)\\ \qquad-L(b_1,\ldots,b_j,d_1,\ldots,d_k)?\end{align*}SAB−(A∪B)L(a1,…,ai,b1,…,bj,d1,…,dk)−L(a1,…,ai,d1,…,dk)−L(b1,…,bj,d1,…,dk)?OnceSSSis extended to the smallest subspaceS′SS′, all the basis vectors ofAAAandBBBwill be retained: for example,2a1b1∈S′,a1b1∈S′2a_1b_1\in S,a_1b_1\in S2a1b1∈S′,a1b1∈S′so their differencea1∈S′a_1\in Sa1∈S′. (DeepSeek-V4-Pro first discovered this.) So the smallest subspace containingSSSisS′ABSABS′AB. Picking a viableCCCis impossible.(⇐\Leftarrow⇐) Trivial.12Prove that the union of two subspaces ofVVVis a subspace ofVVVif and only if one of the subspaces is contained in the other.Proof.(⇒\Rightarrow⇒) Let the two subspaces beUUUandWWW. SupposeU⊈WU\not\subseteq WU⊆WandW⊈UW\not\subseteq UW⊆U, i.e.,∃u∈U−W\exist u\in U-W∃u∈U−Wand∃w∈W−U\exist w\in W-U∃w∈W−U. Thenuw∈U∪Wuw\in U\cup Wuw∈U∪W. Without loss of generality, supposeuw∈Uuw\in Uuw∈U. Thenw(uw)−u∈Uw(uw)-u\in Uw(uw)−u∈U, contradiction.(⇐\Leftarrow⇐) Trivial.Exercises 1B6Let∞\infty∞and−∞-\infty−∞denote two distinct objects, neither of which is inR\mathbb RR. Define an addition and scalar multiplication onR∪{∞,−∞}\mathbb R\cup\{\infty,-\infty\}R∪{∞,−∞}as you could guess from the notation. Specifically, the sum and product of two real numbers is as usual, and fort∈Rt\in\mathbb Rt∈Rdefinet∞{−∞if t0,0if t0,∞if t0,t(−∞){∞if t0,0if t0,−∞if t0,t\infty\begin{cases}-\infty\text{if }t0,\\0\text{if }t0,\\\infty\text{if }t0,\end{cases}\qquad t(-\infty)\begin{cases}\infty\text{if }t0,\\0\text{if }t0,\\-\infty\text{if }t0,\end{cases}t∞⎩⎨⎧−∞0∞ift0,ift0,ift0,t(−∞)⎩⎨⎧∞0−∞ift0,ift0,ift0,andt∞∞t∞∞∞,t(−∞)(−∞)t(−∞)(−∞)−∞,∞(−∞)(−∞)∞0.\begin{align*}t\infty\inftyt\infty\infty\infty,\\ t(-\infty)(-\infty)t(-\infty)(-\infty)-\infty,\\ \infty(-\infty)(-\infty)\infty0.\end{align*}t∞t(−∞)∞(−∞)∞t∞∞∞,(−∞)t(−∞)(−∞)−∞,(−∞)∞0.With these operations of addition and scalar multiplication, isR∪{∞,−∞}\mathbb R\cup\{\infty,-\infty\}R∪{∞,−∞}a vector space overR\mathbb RR? Explain.Solution.(∞(−∞))(−∞)0(−∞)−∞(\infty(-\infty))(-\infty)0(-\infty)-\infty(∞(−∞))(−∞)0(−∞)−∞, while∞((−∞)(−∞))∞(−∞)0\infty((-\infty)(-\infty))\infty(-\infty)0∞((−∞)(−∞))∞(−∞)0. So addition is not associative. Linear Algebra Done Right Partial Solutions
Chapter 1Exercises 1C13Prove that the union of three subspaces ofVVVis a subspace ofVVVif and only if one of the subspaces contains the other two.Proof.(⇒\Rightarrow⇒) Let the three subspaces beA,B,CA,B,CA,B,C. There are two cases:A⊆BA\subseteq BA⊆B. By the result of Exercise 12, eitherB⊆CB\subseteq CB⊆CorC⊆BC\subseteq BC⊆B, in both cases we are done.A⊈B∧B⊈AA\not\subseteq B\land B\not\subseteq AA⊆B∧B⊆A. LetAL(a1,…,ai,d1,…,dk),BL(b1,…,bj,d1,…,dk)AL(a_1,\ldots,a_i,d_1,\ldots,d_k),BL(b_1,\ldots,b_j,d_1,\ldots,d_k)AL(a1,…,ai,d1,…,dk),BL(b1,…,bj,d1,…,dk)wherei0,j0,k≥0i0,j0,k\ge0i0,j0,k≥0. Soa1∈A−B,b1∈B−Aa_1\in A-B,b_1\in B-Aa1∈A−B,b1∈B−A. The smallest subspace containing bothAAAandBBBisABABAB. By the result of Exercise 12, eitherAB⊆CAB\subseteq CAB⊆C(done) orC⊊ABC\subsetneq ABC⊊AB. In the latter case, the question is whetherC⊇SC\supseteq SC⊇S, whereSAB−(A∪B)L(a1,…,ai,b1,…,bj,d1,…,dk)−L(a1,…,ai,d1,…,dk)−L(b1,…,bj,d1,…,dk)?\begin{align*}SAB-(A\cup B)\\ L(a_1,\ldots,a_i,b_1,\ldots,b_j,d_1,\ldots,d_k)\\ \qquad-L(a_1,\ldots,a_i,d_1,\ldots,d_k)\\ \qquad-L(b_1,\ldots,b_j,d_1,\ldots,d_k)?\end{align*}SAB−(A∪B)L(a1,…,ai,b1,…,bj,d1,…,dk)−L(a1,…,ai,d1,…,dk)−L(b1,…,bj,d1,…,dk)?OnceSSSis extended to the smallest subspaceS′SS′, all the basis vectors ofAAAandBBBwill be retained: for example,2a1b1∈S′,a1b1∈S′2a_1b_1\in S,a_1b_1\in S2a1b1∈S′,a1b1∈S′so their differencea1∈S′a_1\in Sa1∈S′. (DeepSeek-V4-Pro first discovered this.) So the smallest subspace containingSSSisS′ABSABS′AB. Picking a viableCCCis impossible.(⇐\Leftarrow⇐) Trivial.12Prove that the union of two subspaces ofVVVis a subspace ofVVVif and only if one of the subspaces is contained in the other.Proof.(⇒\Rightarrow⇒) Let the two subspaces beUUUandWWW. SupposeU⊈WU\not\subseteq WU⊆WandW⊈UW\not\subseteq UW⊆U, i.e.,∃u∈U−W\exist u\in U-W∃u∈U−Wand∃w∈W−U\exist w\in W-U∃w∈W−U. Thenuw∈U∪Wuw\in U\cup Wuw∈U∪W. Without loss of generality, supposeuw∈Uuw\in Uuw∈U. Thenw(uw)−u∈Uw(uw)-u\in Uw(uw)−u∈U, contradiction.(⇐\Leftarrow⇐) Trivial.Exercises 1B6Let∞\infty∞and−∞-\infty−∞denote two distinct objects, neither of which is inR\mathbb RR. Define an addition and scalar multiplication onR∪{∞,−∞}\mathbb R\cup\{\infty,-\infty\}R∪{∞,−∞}as you could guess from the notation. Specifically, the sum and product of two real numbers is as usual, and fort∈Rt\in\mathbb Rt∈Rdefinet∞{−∞if t0,0if t0,∞if t0,t(−∞){∞if t0,0if t0,−∞if t0,t\infty\begin{cases}-\infty\text{if }t0,\\0\text{if }t0,\\\infty\text{if }t0,\end{cases}\qquad t(-\infty)\begin{cases}\infty\text{if }t0,\\0\text{if }t0,\\-\infty\text{if }t0,\end{cases}t∞⎩⎨⎧−∞0∞ift0,ift0,ift0,t(−∞)⎩⎨⎧∞0−∞ift0,ift0,ift0,andt∞∞t∞∞∞,t(−∞)(−∞)t(−∞)(−∞)−∞,∞(−∞)(−∞)∞0.\begin{align*}t\infty\inftyt\infty\infty\infty,\\ t(-\infty)(-\infty)t(-\infty)(-\infty)-\infty,\\ \infty(-\infty)(-\infty)\infty0.\end{align*}t∞t(−∞)∞(−∞)∞t∞∞∞,(−∞)t(−∞)(−∞)−∞,(−∞)∞0.With these operations of addition and scalar multiplication, isR∪{∞,−∞}\mathbb R\cup\{\infty,-\infty\}R∪{∞,−∞}a vector space overR\mathbb RR? Explain.Solution.(∞(−∞))(−∞)0(−∞)−∞(\infty(-\infty))(-\infty)0(-\infty)-\infty(∞(−∞))(−∞)0(−∞)−∞, while∞((−∞)(−∞))∞(−∞)0\infty((-\infty)(-\infty))\infty(-\infty)0∞((−∞)(−∞))∞(−∞)0. So addition is not associative.