1. Solution: (a) $\{(x_1,x_2,x_3)\in\mathbb F^3:x_1+2x_2+3x_3=0\}$ is a subspace of $\mathbb F^3$. By 1.34, to show a subset is a subspace, we just need to check Additive identity, Closed under addition and Closed under scalar multiplication.
Additive identity: it is clear that the additive identity $(0,0,0)$ of $\mathbb F^3$ is contained in $\{(x_1,x_2,x_3)\in\mathbb F^3:x_1+2x_2+3x_3=0\}$.
Closed under addition: if $(a_1,a_2,a_3),(b_1,b_2,b_3)\in \{(x_1,x_2,x_3)\in\mathbb F^3:x_1+2x_2+3x_3=0\}$, then\[a_1+2a_2+3a_3=0\quad\text{and} \quad b_1+2b_2+3b_3=0.\]Hence\[(a_1+b_1)+2(a_2+b_2)+3(a_3+b_3)=(a_1+2a_2+3a_3)+(b_1+2b_2+3b_3)=0,\]this means $(a_1+b_1,a_2+b_2,a_3+b_3)=(a_1,a_2,a_3)+(b_1,b_2,b_3)$ is also in $\{(x_1,x_2,x_3)\in\mathbb F^3:x_1+2x_2+3x_3=0\}$.
Closed under scalar multiplication: if $(a_1,a_2,a_3)\in \{(x_1,x_2,x_3)\in\mathbb F^3:x_1+2x_2+3x_3=0\}$, then $a_1+2a_2+3a_3=0$. For any $\lambda\in\mathbb F$, we have\[\lambda a_1+2(\lambda a_2)+3(\lambda a_3)=\lambda(a_1+2a_2+3a_3)=0.\]This means \[\lambda(a_1,a_2,a_3)=(\lambda a_1,\lambda a_2,\lambda a_3)\in \{(x_1,x_2,x_3)\in\mathbb F^3:x_1+2x_2+3x_3=0\}.\]
(b) $\{(x_1,x_2,x_3)\in\mathbb F^3:x_1+2x_2+3x_3=4\}$ is not a subspace of $\mathbb F^3$ since $(0,0,0)$ is not in it.
(c) $\{(x_1,x_2,x_3)\in\mathbb F^3:x_1 x_2 x_3=0\}$ is not a subspace of $\mathbb F^3$ since $(1,1,0)$ and $(0,1,1)$ are in it, but the sum $(1,2,1)=(1,1,0)+(0,1,1)$ is not in it.
(d) $\{(x_1,x_2,x_3)\in\mathbb F^3:x_1=5x_3\}$ is a subspace of $\mathbb F^3$.
Additive identity: it is clear that the additive identity $(0,0,0)$ of $\mathbb F^3$ is contained in $\{(x_1,x_2,x_3)\in\mathbb F^3:x_1=5x_3\}$.
Closed under addition: if $(a_1,a_2,a_3),(b_1,b_2,b_3)\in \{(x_1,x_2,x_3)\in\mathbb F^3:x_1=5x_3\}$, then $a_1=5a_3$ and $b_1=5b_3$. Hence\[a_1+b_1=5a_3+5b_3=5(a_3+b_3),\]this means $(a_1+b_1,a_2+b_2,a_3+b_3)=(a_1,a_2,a_3)+(b_1,b_2,b_3)$ is also in $\{(x_1,x_2,x_3)\in\mathbb F^3:x_1=5x_3\}$.
Closed under scalar multiplication: if $(a_1,a_2,a_3)\in \{(x_1,x_2,x_3)\in\mathbb F^3:x_1=5x_3\}$, then $a_1=5a_3$. For any $\lambda\in\mathbb F$, we have $\lambda a_1=\lambda(5 a_3)=5(\lambda a_3)$. This means \[\lambda(a_1,a_2,a_3)=(\lambda a_1,\lambda a_2,\lambda a_3)\in \{(x_1,x_2,x_3)\in\mathbb F^3:x_1=5x_3\}.\]
2. Solution: (a) If this set is a subspace of $\mathbb F^4$, then $(0,0,0,0)\in \mathbb F^4$, then $0=5\cdot 0+b$. Hence $b=0$. Follow the same steps of Problem 1 (d), we will see the set is a subspace of $\mathbb F^4$ if $b=0$.
(b) (c) and (d) is similar to Problem 3 and 4.
Now let us consider (e). Denote the set of all sequences of complex numbers with limit $0$ by $A$.
Additive identity: it is clear that $(0,0,\cdots)\in A$.
Closed under addition: if $(a_1,a_2,\cdots),(b_1,b_2,\cdots)\in A$, then\[\lim_{n\to\infty}a_n=0\quad\text{and}\quad\lim_{n\to\infty}b_n=0.\]It is easy to see\[\lim_{n\to\infty}(a_n+b_n)=\lim_{n\to\infty}a_n+\lim_{n\to\infty}b_n=0+0=0.\]This means $(a_1+b_1,a_2+b_2,\cdots)=(a_1,a_2,\cdots)+(b_1,b_2,\cdots)\in A$.
Closed under scalar multiplication: if $(a_1,a_2,\cdots)\in A$, then\[\lim_{n\to\infty}a_n=0.\]For any $\lambda\in\mathbb C$, it is easy to see\[\lim_{n\to\infty}(\lambda a_n)=\lambda\lim_{n\to\infty} a_n=\lambda 0=0.\]This means $\lambda(a_1,a_2,\cdots)=(\lambda a_1,\lambda a_2,\cdots)\in A$.
3. Solution: Denote the set of differentiable real-valued functions $f$ on the interval $(-4,4)$ such that $f'(-1)=3f(2)$ by $V$.
Additive identity: it is clear that the constant function $f\equiv 0$ is contained in $V$.
Closed under addition: if $f,g\in V$, then $f$ and $g$ are differentiable real-valued functions. So is $f+g$. Moreover,\[(f+g)'(-1)=f'(-1)+g'(-1)=3f(2)+3g(2)=3(f(2)+g(2))=3(f+g)(2).\]This concludes $V$ is closed under addition.
Closed under scalar multiplication: if $f\in V$, for any $\lambda\in \mathbb R$, then $f$ is differentiable real-valued functions. So is $\lambda f$. Moreover,\[(\lambda f)'(-1)=\lambda f'(-1)=\lambda (3f)(2)=3(\lambda f)(2).\]This deduces $V$ is closed under scalar multiplication.
4. Solution: Denote the set of continuous real-valued functions $f$ on the interval $[0,1]$ such that $\int_0^1f=b$ by $V_b$.
If $V_b$ is a subspace of $\mathbb R^{[0,1]}$, then for any $f\in V_b$, we have $\int_0^1f=b$. Because $V_b$ is a subspace of $\mathbb R^n$, it follows that $kf\in V_b$ for any $k\in\mathbb R$. Hence\[b=\int_0^1(kf)=k\int_0^1f=kb,\quad \text{for all}~k\in\mathbb R,\]this happens if and only if $b=0$.
Now if $b=0$, then for any $f,g\in V_0$ and $\lambda\in\mathbb R$. We have that\[\int_0^1(f+g)=\int_0^1f+\int_0^1g=0+0=0\]and $f+g$ is continuous real-valued functions since $f$ and $g$ are. This deduces $f+g\in V_0$, i.e. $V_0$ is closed under addition. Similarly,\[\int_0^1(\lambda f)=\lambda\int_0^1f=k0=0\]and $\lambda f$ is continuous real-valued functions since $f$ is. This implies $\lambda f\in V_0$, i.e. $V_0$ is closed under scalar multiplication. On the other hand, the constant function $f\equiv 0\in V_0$, which is also the additive identity in $\mathbb R^{[0,1]}$. Hence $V_0$ is a subspace of $\mathbb R^n$ by 1.34.
5. Solution: Note that we consider complex vector space, so if $\mathbb R^2$ is a subspace of the complex vector space $\mathbb C^2$, then\[i(1,1)=(i,i)\in\mathbb R^2,\]we get a contradiction. Hence $\mathbb R^2$ is not a subspace of the complex vector space $\mathbb C^2$.
6. Solution: (a) Because $a^3=b^3$ if and only if $a=b$ in $\mathbb R$, hence \[\{(a,b,c)\in\mathbb R^3:a^3=b^3\}=\{(a,b,c)\in\mathbb R^3:a=b\}\]is obviously a subspace of $\mathbb R^3$ by the similar arguments in Problem 1 and Problem 2.
(b) Note that\[x=\left(1,\frac{-1+\sqrt{3}i}{2},0\right)\in\{(a,b,c)\in\mathbb C^3:a^3=b^3\}\]and\[y=\left(1,\frac{-1-\sqrt{3}i}{2},0\right)\in\{(a,b,c)\in\mathbb C^3:a^3=b^3\}.\]However,\[x+y=(2,-1,0)\notin \{(a,b,c)\in\mathbb C^3:a^3=b^3\}.\]This implies $\{(a,b,c)\in\mathbb C^3:a^3=b^3\}$ is not closed under addition, hence not a subspace of $\mathbb C^3$.
7. Solution: Denote $\{(x,y)\in\mathbb R^2:x,y\in\mathbb Z\}$ by $U$, then $U$ is not empty. If $(x_1,y_1)\in A$ and $(x_2,y_2)\in A$, then $x_1,x_2,y_1,y_2\in\mathbb Z$. Hence $x_1+x_2$ and $y_1+y_2$ are integers. This means $(x_1+x_2,y_1+y_2)=(x_1,y_1)+(x_2,y_2)\in U$, i.e. $U$ is closed under addition. Similarly, since $(-x_1,-y_1)\in U$, it follows that $U$ is closed under additive inverses. However, $U$ is not closed under scalar multiplication since $(1,1)\in A$ while $\dfrac{1}{2}(1,1)\notin U$. Hence $U$ is not a subspace of $\mathbb R^2$.
8. Solution: Denote $\{(x,y)\in \mathbb R^2: x=0~\text{or}~y=0\}$ by $U$, then $U$ is not empty. If $(x,0)\in U$, then for any $\lambda\in\mathbb R$, we have\[\lambda(x,0)=(\lambda x,0)\in U.\]Similarly, $\lambda(0,y)\in U$, hence $U$ is closed under scalar multiplication. However, $(1,0),(0,1)\in U$ while $(1,1)=(1,0)+(0,1)\notin U$. This implies $U$ is not closed under addition, hence not a subspace of $\mathbb R^2$.
9. Solution: Denote the set of periodic functions from $\mathbb R$ to $\mathbb R$ by $S$. Then $S$ is not a subspace of $\mathbb R^{\mathbb R}$. Otherwise, we have $h(x)=\sin\sqrt{2}x+\cos x$ since both $f(x)=\sin\sqrt{2}x$ and $g(x)=\cos x$ are periodic functions from $\mathbb R$ to $\mathbb R$. Assume there exists a positive number $p$ such that $h(x)=h(x+p)$ for all $x\in\mathbb R$, then $1=h(0)=h(p)=h(-p)$. It is equivalent to \[1=\cos p+\sin\sqrt{2}p=\cos p-\sin\sqrt{2}p,\]this implies $\sin\sqrt{2}p=0$ and $\cos p=1$. $\cos p=1$ deduces that $p=2k\pi$ where $k\in \mathbb Z$. However $\sin\sqrt{2}p=0$ implies $\sqrt{2}p=l\pi$ where $l\in\mathbb Z$. Hence \[\sqrt{2}=\frac{l\pi}{2k\pi}=\frac{l}{2k}\in\mathbb Q,\]which is impossible. Therefore we get the conclusion.
10. Solution: Additive identity: by definition $0\in U_1$ and $0\in U_2$, hence $0\in U_1\cap U_2$.
Closed under addition: if $x\in U_1\cap U_2$ and $y\in U_1\cap U_2$, then $x\in U_1$ and $y\in U_1$, hence $x+y\in U_1$ for $U_1$ is closed under addition. Similarly, $x+y\in U_2$. Therefore $x+y\in U_1\cap U_2$.
Closed under scalar multiplication: if $x\in U_1\cap U_2$, then $x\in U_1$. Then for any $\lambda\in\mathbb F$, we have $\lambda x\in U_1$ since $U_1$ is closed under scalar multiplication. Similarly, $\lambda x\in U_2$. Therefore $\lambda x\in U_1\cap U_2$.
11. Solution: Assume $U_i$ are subspaces of $V$, where $i\in I$. Now we will show $\cap_{i\in I}U_i$ is a subspace of $V$.
Additive identity: by definition $0\in U_i$ for every $i\in I$, hence $0\in \cap_{i\in I}U_i$.
Closed under addition: if $x\in \cap_{i\in I}U_i$ and $y\in \cap_{i\in I}U_i$, then for any given $i\in I$, we have $x\in U_i$ and $y\in U_i$, hence $x+y\in U_i$ for $U_i$ is closed under addition. Therefore $x+y\in \cap_{i\in I}U_i$.
Closed under scalar multiplication: if $x\in \cap_{i\in I}U_i$, then $x\in U_i$ for any given $i\in I$. Then for any $\lambda\in\mathbb F$, we have $\lambda x\in U_i$ since $U_i$ is closed under scalar multiplication. Therefore $\lambda x\in \cap_{i\in I}U_i$.
12. Solution: Suppose $U$ and $W$ are two subspaces of $V$. We argue it by contradiction. If $U\cup W$ is a subspace of $V$, moreover $U\nsubseteq W$ and $W\nsubseteq U$. Consider $u\in U\setminus W$ and $w\in W\setminus U$, then $u+w\in U\cup W$ since $U\cup W$ is a subspace of $V$. Hence $u+w\in U$ or $W$. If $u+w\in U$, then $w=(u+w)-u\in U$. We get a contradiction. If $u+w\in W$, then $u=(u+w)-w\in W$. We also get a contradiction. Hence if $U\cup W$ is a subspace of $V$, we must have $U\subset W$ or $W\subset U$.
If $U\subset W$ or $W\subset U$. Without loss of generality, we can assume $U\subset W$. Then $U\cup W=W$ is obviously a subspace of $V$.
13. See If a field $\mb F$ is such that $|\mb F|>n−1|$ why is $V$ a vector space over $\mb F$ not equal to the union of proper subspaces of $V$ or A question on vector space over an infinite field.
14. Solution: It is clear that $U$ and $W$ are subspaces of $\mathbb{F}^4$. Now assume that $(x_1,x_1,y_1,y_1)\in U$ and $(x_2,x_2,x_2,y_2)\in W$, then \begin{align*} &(x_1,x_1,y_1,y_1)+(x_2,x_2,x_2,y_2)\\=&(x_1+x_2,x_1+x_2,y_1+x_2,y_1+y_2) \in \{(x,x,y,z):x,y,z\in\mathbb F\}. \end{align*} Hence $U+W\subset \{(x,x,y,z):x,y,z\in\mathbb F^4\}$. For any $x,y,z\in\mathbb F$, we have $(0,0,y-x,y-x)\in U$ and $(x,x,x,z+x-y)\in W$. However, \[ (x,x,y,z)=(0,0,y-x,y-x)+(x,x,x,z+x-y)\in U+W, \]hence $\{(x,x,y,z):x,y,z\in\mathbb F^4\}\subset U+W$. Combining this with previous argument, it follows that $U+W=\{(x,x,y,z):x,y,z\in\mathbb F\}$.
15. Solution: Because $U$ is a subspace of $V$, hence closed under addition.Therefore for any $x,y\in U$, we have $x+y\in U$, i.e. $U+U\subset U$. Note that if $x\in U$, then $x=x+0\in U+U$, hence $U\subset U+U$. Combining with these arguments, it follows that $U+U=U$.
16. Solution: For $x\in U$ and $y\in W$, because addition in V is commutative, we have $x+y=y+x\in W+U$. This implies $U+W\subset W+U$. Similarly, we have $W+U\subset U+W$. Hence $U+W=W+U$.
17. Solution: Note that in V, we have $(x+y)+z=x+(y+z)$. Hence this is similar to Problem 16. Let $x_i\in U_i$, $i=1,2,3$, then \[(x_1+x_2)+x_3=x_1+(x_2+x_3)\in U_1+(U_2+U_3).\]Since every element in $(U_1+U_2)+U_3$ can be expressed as the form $(x_1+x_2)+x_3$, it follows that $(U_1+U_2)+U_3\subset U_1+(U_2+U_3)$. Similarly, we also have $U_1+(U_2+U_3)\subset (U_1+U_2)+U_3$. Hence $(U_1+U_2)+U_3=U_1+(U_2+U_3)$.
18. Solution: If $U$ is a additive identity, then for any subspace $W$ of $V$, we have $U+W=W$. This means $U\subset W$ by the similar arguments in Problem 15 and 16. Hence the only possibility is $U=0$, in fact this is the case. Hence $0$ is the additive identity. Suppose subspace $W$ of $V$ has additive inverses, then there exists a subspace $S$ of $V$ such that $W+S=0$. This can only happen when $W=0$ since $W\subset W+S$.
19. Solution: Here is a counterexample. Let $V=U_1=\{(a,b)\in\R^2:a,b\in\R\}$, $U_2=\{(a,0)\in\R^2:a\in\R\}$ and $W=\{(0,b)\in\R^2:b\in\R\}$. Then it is easy to see that $U_1+W=U_2+W$, but $U_1\ne U_2$.
See comments for the example from Neven Sajko.
20. Solution: Take $W=\{(0,z,0,w)\in\mb F^4:z,w\in\mb F\}$. For any $(x,y,z,w)\in\mb F^4$, we have\[(x,y,z,w)=(x,x,z,z)+(0,y-x,0,w-z)\in U+W\]since $(x,x,z,z)\in U$ and $(0,y-x,0,w-z)\in W$. We have $\mb F^4=U+W$.
Moreover, if $(x,y,z,w)\in U\cap W$, then we must have $x=y$ and $z=w$ since $(x,y,z,w)\in U$.
Similarly, since $(x,y,z,w)\in W$, we have $x=0$ and $z=0$. Therefore, $x=y=0$ and $z=w=0$, hence $(x,y,z,w)=(0,0,0,0)$. It follows that $U\cap W=\{0\}$. Hence $F=U\oplus W$ by 1.45.
21. Solution: Since we can only choose the first two coordinates arbitrarily in $U$, we can take $W$ in which the first two coordinates are zero and the last three are variables.
Let $W=\{(0,0,z,w,s)\in\mb F^5:x,y\in\mb F\}$. If $(x,y,z,w,s)\in U\cap W$, then $x=y=0$ since $(x,y,z,w,s)\in W$. Moreover, we have $z=x+y,w=x-y,s=2x$ since $(x,y,z,w,s)\in U$. Therefore $z=w=s=0$, hence $U\cap W=\{0\}$.
Again, for any $(x,y,z,w,s)\in\mb F^5$, we have\[(x,y,z,w,s)=(x,y,x+y,x-y,2x)+(0,0,z-x-y,w-x+y,s-2x)\in U+W\]since $(x,y,x+y,x-y,2x)\in U$ and $(0,0,z-x-y,w-x+y,s-2x)\in W$. Therefore $\mb F^5=U+W$.
Since $U\cap W=0$, it follows from 1.45 that $\mb F^5 =U\oplus W$.
22. Solution: Let $W_1=\{(0,0,z,0,0)\in\mb F^5:x,y\in\mb F\}$, $W_2=\{(0,0,0,z,0)\in\mb F^5:x,y\in\mb F\}$ and $W_3=\{(0,0,0,0,z)\in\mb F^5:x,y\in\mb F\}$. By the same argument as in Problem 21, we have\[\mb F^5 =U\oplus W_1\oplus W_2\oplus W_3.\]
23. Solution: Here is a counter example. Let $V=\R^2$, $U_1=\{(x,0)\in\R^2:x\in\R\}$, $U_2=\{(0,y)\in\R^2:y\in\R\}$ and $W=\{(z,z)\in\R^2:z\in\R\}$. By the same argument as in Problem 21, one has\[V=U_1\oplus W \quad\text{and}\quad V=U_2\oplus W,\]however $U_1\ne U_2$.
This is exactly the example from Neven Sajko.
24. Solution: Given any $f\in \R^\R$, define $f_e(x)$ to be $\dfrac{f(x)+f(-x)}{2}$ and define $f_o(x)$ to be $\dfrac{f(x)-f(-x)}{2}$ for all $x\in\R$ . Then $f_e,f_o\in\R^\R$. Moreover, for all $x\in \R$, we have\[f_e(-x)=\frac{f(-x)+f(x)}{2}=\frac{f(x)+f(-x)}{2}=f_e(x)\]and\[f_o(-x)=\frac{f(-x)-f(x)}{2}=-\frac{f(x)-f(-x)}{2}=-f_o(x).\]Hence $f_e\in U_e$ and $f_o\in U_o$. Note that we also have\[f(x)=\frac{f(x)+f(-x)}{2}+\frac{f(x)-f(-x)}{2}=f_e(x)+f_o(x),\]hence $f=f_e+f_o\in U_e+U_o$. Since we can choose $f$ arbitrarily, one has $\R^\R=U_e+U_o$.
By 1.45, to show $\R^\R=U_e\oplus U_o$, it suffices to prove that $U_e\cap U_o=\{0\}$. Let $f\in U_e\cap U_o$, then $f(x)=f(-x)$ since $f\in U_e$ and $f(x)=-f(-x)$ since $f\in U_o$ for all $x\in\R$ . Sum up $f(x)=f(-x)$ and $f(x)=-f(-x)$, we have $f(x)=0$ for all $x\in \R$. Hence $f=0$, which implies $U_e\cap U_o=\{0\}$.
sam
5 Jan 2021I was wondering where the idea for solving 6b comes from, and I remember Axler drops a hint of one of the complex cube roots of 1 in exercise 1.A.2. Clever.
cao
24 Nov 2020About 1.C.3,can the additive identity be the function
with the point f'(-1)=0 and f(2)=0, but others area of [-4, 4] are continuous and deritive ? so this condition are more loose than all the range equals 0!
Jorge Opaso
19 Nov 202023.
U_1=V-W+{0}
U_2=V-W+{0}
Marie Dupont
13 Jun 2020I do not understand the work presented for Q14. Isn't it enough to show that the addition of any two arbitrary elements (one belonging to U and the other to W) is equal to the set {(x,x,y,z) 'element' F^4 : x,y,z 'element' F} ? In short, doesn't the first part of your argument suffice?
Linearity
15 Jun 2020The first part shows the addition of any two arbitrary elements is of form $(x,x,y,z)$. The second part shows all elements of form $(x,x,y,z)$ can be written as the addition of any two arbitrary elements.
Recall that if you would like to show two sets $A$ and $B$ are equal, then we usually show $A\subset B$ and $B\subset A$. That is why we have two parts here.
Marie
26 Jun 2020thank you so much!
Tianze
21 May 2020I seem to have found a mistake. In question 14, it should be written like this"$x,y,z \in \mathbb{F}$", but you wrote"$x,y,z \in \mathbb{F}^4$".Right?
Benjamin B
16 Aug 2019I don't quite follow the logic in #12. You say $u + w \in U \cup W$ since $U \cup W$ is a subspace of $V$, but $u + w$ is in $U \cup W$ by construction, not because $U \cup W$ is a subspace of V. Then, I'm not sure how it's asserted that $u + w$ must be in $U$ or $W$.
Benjamin B
16 Aug 2019Oops, I misread. Got it now!
Abhinav Patil
4 Mar 2019For 19: The examples given by @nevensajko:disqus seem needlessly complicated. Rather than making W the identity function (x = y), make W the entire xy plane. Then U1 (x-axis) and U2 (y-axis) are obviously not equal, but as each is a subset of the plane, U1 + W = W and U2 + W = W which means U1 + W = U2 + W
Hanson Char
11 Feb 2018Exercise 24 means any function in R^R can be decomposed into the sum of two other functions, one odd and one even. Amazing!
Hanson Char
9 Feb 2018Another solution for 8:
$$\left\{(x, y) \in \mathbb{R}^{2} :|x|=|y|\right\}$$
Hanson Char
8 Feb 2018Another possible solution: $\left\{(x, y, z) \in \mathbb{R}^{2} \backslash(0,0,0)\right\}$
Tianze
21 May 2020However, since you have an additive inverse, you must have an addiitive identity
Hanson Char
8 Feb 20187 Another solution: $\{(x,y,z) \in \mathbb{R}^2 \setminus(0,0,0) \}$
Photon Guo
1 Sep 2019if $u,-u\in U$,then $0= u+(-u)$ must be in $U$, otherwise it is not closed under addition.
Brian Nguyen
19 Jan 2018I don't agree #14). It appears as though the second part of your argument where (x,x,x,z+x-y) ∈ W. I do not think this is valid. The (0,0,y-x,y-x) ∈ U is true if you use the closure by addition.
Hanson Char
10 Feb 2018Given W = {(x, x, x, y) ∈ F^4: x, y ∈ F}, for any x, y, z ∈ F^4, (x, x, x, z+x-y) ∈ W.
Brian Nguyen
14 Jan 20186) b) doesn't seem right. The vectors you chosen for x and y does not satisfy the condition a^3 = b^3; that condition isn't correct, the proof appears wrong and the answer should be that {(a,b,c) C^3: a^3 = b^3} is indeed a subspace of C^3
From the book: Definition of Subspace
A subset U of V is a subspace of V if and only if U satisfies the following
three conditions:
1) 0 ∈ U
2) u,w ∈ U
3) a ∈ F, u ∈ U implies au ∈ U
where F can be real or complex vector space.
since you're trying to say that x,y ∈ {(a,b,c), C^3: a^3 = b^3} and the vectors you chose are not part of the set. So, it's not right.
rage2050a
13 Dec 2017In 9, it is written "1 = cos p + sin sqrt2 p = cos p - sin sqrt2 p
this implies $\sin \sqrt{2 p} = 0$ and $\cos p = 1$"
but it is not necessarily so.
Numerically, p=1.45459662, sin sqrt2 p = 0.884061615 (not 0), cos p = 0.115938388 (not 1), is essentally 1.
Hanson Char
9 Feb 2018Hanson Char
9 Feb 2018Hanson Char
9 Feb 2018btw, solution 9 is a very clever construction of a counter example.
Nate Brugnone
3 Jul 2017For #11, showing that the intersection is closed under scalar multiplication we have: $x \in \cap_{i \in I} U_i \Rightarrow x \in U_i$. This is fine, but how do we suppose that $\lambda x \in U_i \Rightarrow \lambda x \in \cap_{i \in I} U_i$? Isn't it entire plausible that $\lambda x \in U_i$, but $\lambda x \notin \cap_{i \in I} U_i$?
Nate Brugnone
3 Jul 2017Hmm, LaTeX doesn't appear to be supported any longer. So, my question is essentially: How does k*x ∈ U_i imply that k*x ∈ intersection of U_i's? Isn't it entirely plausible that x ∈ intersection of U_i's but that k*x is not ∈ intersection of U_i's (understanding that k*x ∈ U_i by definition)?
Photon Guo
1 Sep 2019if $x\in U_i$, then $kx\in U_i$ because $U_i$ is closed under scalar multiplication. similarily, if $x$ is the element of the union of $U_i$, then $x\in U_i$ ($1\leq i\leq n$), so $kx\in U_i$ ($1\leq i\leq n$), which means $kx$ is also the element of the union.
Neven Sajko
6 Feb 201624
Space of real functions: R^R.
Uₑ= {f∈R^R : f(-x)=f(x)} (The set of even functions.)
Uₒ= {f∈R^R : f(-x)=-f(x)} (The set of odd functions.)
Show that R^R = Uₑ⊕Uₒ.
Proof:
Firstly we prove Uₑ,Uₒ≤R^R.
0∈Uₑ; ∀x∈R 0(x)=0; 0(-x)=0=0(x).
f,g∈Uₑ → (f+g)(-x) = f(-x) + g(-x) = f(x) + g(x) = (f+g)(x).
g∈Uₑ, λ∈R → (λg)(-x) = λg(-x) = λg(x) = (λg)(x).
Similarly Uₒ≤R^R.
We need to prove Uₑ∩Uₒ = {0}.
f∈Uₑ∩Uₒ → f(x) = f(-x) = -f(x) → f=-f → f=0.
Now we only need to prove Uₑ+Uₒ=R^R.
Uₑ+Uₒ⊆R because Uₑ,Uₒ⊂R^R and R^R is closed under addition.
Uₑ+Uₒ⊇R because ∀f∈R^R ∀x∈R f(x) = ½f(x) + ½f(x) + ½f(-x) -½f(-x) = ½(f(x)+f(-x)) + ½(f(x)-f(-x)) where the first summand is from Uₑ and the second from Uₒ.
QED
Neven Sajko
2 Feb 201623
The counter example from 19 still works.
jassie
14 Jun 2020No it doesnt work, for this is direct sum and so the answer is wrong.
For a direct sum a 0 can be written in only one way.
Your wrong solution allows it to be written in two ways, for example for W+U2:
(0,0)+(0,0) and also (0,-4)+(0,4)=(0,0) (and infinite others).
I think this exercise indeed has no counterexample and it can be proven.
Neven Sajko
2 Feb 201622
Just decompose W from exercise 21 into 3 parts.
Neven Sajko
2 Feb 201621
W = {(0,0,a,b,c)∈F⁵ : a,b,c∈F}
Neven Sajko
1 Feb 201620
W = {(0,x,y,0) : x,y∈F}
JGer
29 Mar 2016Can I ask why this is the case? It would appear to me that, using the W you've provided, that U+W=(x,x,y,y), which, doesn't seem to me to be all of F^4.
Xianhe Yangzhang
11 May 2016I think it should be W={ (0,y,x,0) : x,y in F}.
Eduardo Furihata
8 Jun 2016consider U = {(x1,x1,y1,y1):x1,y1 ∈ F}
and W = {(0,x2,y2,0) : x2,y2∈F}
F^4 = U+W = {(x1, x1+x2, y1+y2, y1) : x1, y1, x2, y2 ∈ F}
Hanson Char
11 Feb 2018Indeed, it works if W is one of the four subspaces:
{(0, x, 0, y) : x, y ∈ F}
{(0, x, y, 0) : x, y ∈ F}
{(x, 0, 0, y) : x, y ∈ F}
{(x, 0, y, 0) : x, y ∈ F}
and many more.
Faris
3 Aug 2019How is it so when U ∩ W is {(0,x,y,0) : x,y ∈ F} and not {0} (correct me if I'm wrong)?
Neven Sajko
1 Feb 201619
A counter example:
U₁= {λ(1,0) : λ∈F} (The x axis.)
U₂= {λ(0,1) : λ∈F} (The y axis.)
W = {λ(1,1) : λ∈F} (The x=y line/the identity function graph.)
Obviously U₁≠U₂, but:
U₁+ W = {a(1,0)+b(1,1) : a,b∈F}
= {(a(1,0)-b(1,0))+b(1,1) : a,b∈F} (Because U₁ is closed under addition.)
= {a(1,0)+b(0,1) : a,b∈F} (Note that this is the x,y plane.)
= {a(1,0)+(a(0,1)+b(0,1)) : a,b∈F}
= {a(1,1)+b(0,1) : a,b∈F}
= W + U₂ = U₂+ W
Luis
13 Nov 2020I don't see clear how you do the step on the second equality of U_1+W.
I understand that U_1 is closed under addition because it is a vectorspace but why that imply that doesn't mean that a(1,0) = a(1,0)-b(1,0)
Linearity
16 Nov 2020Because $a$ can be any number in $\mathbb F$. You can think it as \[a(1,0)=\tilde a(1,0)-b(1,0)\]where $\tilde a=a+b$ and then replace $\tilde a$ with $a$.