06-240/Term Test: Difference between revisions
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'''Problem 2.''' |
'''Problem 2.''' |
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# In the field <math>{\mathbb C}</math> of complex numbers, compute <center><math>\frac{1}{2 |
# In the field <math>{\mathbb C}</math> of complex numbers, compute <center><math>\frac{1}{2+3i}+\frac{1}{2-3i}</math> and <math>\frac{1}{2+3i}-\frac{1}{2-3i}</math>.</center> |
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# Working in the field <math>{\mathbb Z}/7</math> of integers modulo 7, make a table showing the values of <math>a^{-1}</math> for every <math>a\neq 0</math>. |
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'''Problem 3.''' Let <math>V</math> be a vector space and let <math>W_1</math> and <math>W_2</math> be subspaces of <math>V</math>. Prove that <math>W_1\cup W_2</math> is a subspace of <math>V</math> iff <math>W_1\subset W_2</math> or <math>W_2\subset W_1</math>. |
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'''Problem 4.''' In the vector space <math>M_{2\times 2}({\mathbb Q})</math>, decide if the matrix <math>\begin{pmatrix}1&2\\-3&4\end{pmatrix}</math> is a linear combination of the elements of <math>S=\left\{\begin{pmatrix}1&0\\-1&0\end{pmatrix},\ \begin{pmatrix}0&1\\0&1\end{pmatrix},\ \begin{pmatrix}1&1\\0&0\end{pmatrix}\right\}</math>. |
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'''Problem 5.''' Let <math>V</math> be a finite dimensional vector space and let <math>W_1</math> and <math>W_2</math> be subspaces of <math>V</math> for which <math>W_1\cap W_2=\{0\}</math>. Denote the linear span of <math>W_1\cup W_2</math> by <math>W_1+W_2</math>. Prove that <math>\dim(W_1+W_2)=\dim W_1 + \dim W_2</math>. |
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<center> '''Good Luck!''' </center> |
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{{06-240/Results of the Term Test}} |
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==Solution Set== |
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Students are most welcome to post a solution set here. |
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Solutions posted by me, feel free to make any changes: |
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'''Problem 1.''' <math>0_F\cdot v=(0_F+0_F)\cdot v</math> (by F3) |
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<math>(0_F+0_F)\cdot v=0_F\cdot v+0_F\cdot v</math> (by VS8) |
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By VS4, <math>\exists\ (0_F\cdot v)' s.t. (0_F\cdot v)+(0_F\cdot v)'=0_V</math> |
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Add <math>(0_F\cdot v)'</math> to both sides of <math>0_F\cdot v=0_F\cdot v+0_F\cdot v</math> |
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<math>(0_F\cdot v)'+(0_F\cdot v)=[(0_F\cdot v)'+0_F\cdot v]+0_F\cdot v</math> |
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<math>0_V=0_V+0_F\cdot v</math> (by construction) |
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<math>0_V=0_F\cdot v</math> (by VS3) |
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'''Problem 2.''' 1) <math>\frac{1}{2+3i}+\frac{1}{2-3i}=\frac{(2-3i)+(2+3i)}{(2+3i)(2-3i)}=\frac{4}{2^2+3^3}=\frac{4}{13}</math> |
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<math>\frac{1}{2+3i}-\frac{1}{2-3i}=\frac{(2-3i)-(2+3i)}{(2+3i)(2-3i)}=\frac{-6i}{2^2+3^3}=-\frac{6}{13}i</math> |
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2) <math>1^{-1}=1</math> |
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<math>2^{-1}=4</math> |
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<math>3^{-1}=5</math> |
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<math>4^{-1}=2</math> |
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<math>5^{-1}=3</math> |
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<math>6^{-1}=6</math> |
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'''Problem 3.''' First suppose <math>W_1\subset W_2</math> or <math>W_2\subset W_1</math>, then <math>W_1\cup W_2</math> equals <math>W_2</math> or <math>W_1</math>. Either case <math>W_1\cup W_2</math> is a subspace of <math>V</math>. |
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Now suppose <math>W_1\cup W_2</math> is a subspace of <math>V</math> and neither <math>W_1\subset W_2</math> nor <math>W_2\subset W_1</math>. It must follows <math>\exists\ x\in W_1</math> s.t. <math>x</math> is not in <math>W_2</math> and <math>\exists\ y\in W_2</math> s.t. <math>y</math> is not in <math>W_1</math>. Since <math>x,y\in W_1\cup W_2</math>, <math>x+y\in W_1\cup W_2</math>. If <math>x+y\in W_1</math>, then <math>\exists\ -x\in W_1</math> s.t. <math>(-x)+x+y\in W_1</math> and <math>(-x)+x=0</math>, so <math>y\in W_1</math>, a contradiction. If <math>x+y\in W_2</math>, then <math>\exists\ -y\in W_2</math> s.t. <math>x+y+(-y)\in W_2</math> and <math>(-y)+y=0</math>, so <math>x\in W_2</math>, a contradiction. Therefore either <math>W_1\subset W_2</math> or <math>W_2\subset W_1</math>. |
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'''Problem 4.''' Suppose <math>\begin{pmatrix}1&2\\-3&4\end{pmatrix}</math> is a linear combination of <math>\begin{pmatrix}1&0\\-1&0\end{pmatrix}</math>, <math>\begin{pmatrix}0&1\\0&1\end{pmatrix}</math>, and <math>\begin{pmatrix}1&1\\0&0\end{pmatrix}</math>, then <math>\exists\ a_1, a_2, a_3\in F</math> s.t. <math>\begin{pmatrix}1&2\\-3&4\end{pmatrix}=a_1\begin{pmatrix}1&0\\-1&0\end{pmatrix}+a_2\begin{pmatrix}0&1\\0&1\end{pmatrix}+a_3\begin{pmatrix}1&1\\0&0\end{pmatrix}</math>. |
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<math>\mbox{Need to solve}\begin{cases} |
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1=a_1^{}+a_3^{}\\ |
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2=a_2^{}+a_3^{}\\ |
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-3=-a_1^{}\\ |
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4=a_2^{}\end{cases}</math> |
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Solving the equations yields <math>a_1=3</math>, <math>a_2=4</math>, <math>a_3=-2</math>, so <math>\begin{pmatrix}1&2\\-3&4\end{pmatrix}</math> is a linear combination of <math>\begin{pmatrix}1&0\\-1&0\end{pmatrix}</math>, <math>\begin{pmatrix}0&1\\0&1\end{pmatrix}</math>, and <math>\begin{pmatrix}1&1\\0&0\end{pmatrix}</math>. Specifically, <math>\begin{pmatrix}1&2\\-3&4\end{pmatrix}=3\begin{pmatrix}1&0\\-1&0\end{pmatrix}+4\begin{pmatrix}0&1\\0&1\end{pmatrix}-2\begin{pmatrix}1&1\\0&0\end{pmatrix}</math>. |
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'''Problem 5.''' Since <math>V</math> is finite-dimensional, then so are <math>W_1</math> and <math>W_2</math> and their basis. Let <math>{a_1, a_2,..., a_m}</math> be the basis of <math>W_1</math> and <math>{b_1, b_2,..., b_n}</math> be the basis of <math>W_2</math> so <math>\dim W_1=m</math> and <math>\dim W_2=n</math>. |
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We know <math>{a_1, a_2,..., a_m}</math> and <math>{b_1, b_2,..., b_n}</math> are linearly independent and clearly <math>{a_1, a_2,..., a_m, b_1, b_2,..., b_n}</math> spans <math>W_1+W_2</math>. If <math>{a_1, a_2,..., a_m, b_1, b_2,..., b_n}</math> is linearly dependent, then there exist not all zero coefficients <math>c_1, c_2,..., c_{m+n}\in F</math> s.t. <math>c_1 a_1+c_2 a_2+...+c_m a_m+c_{m+1} b_1+c_{m+2} b_2+...+c_{m+n} b_n=0</math>. Then some linear combinations of <math>{a_1, a_2,..., a_m}</math> with not all zero coefficients can be expressed in linear combinations of <math>{b_1, b_2,..., b_n}</math>, but this would imply <math>W_1\cap W_2\neq \{0\}</math>, a contradiction. Therefore <math>{a_1, a_2,..., a_m, b_1, b_2,..., b_n}</math> is a linearly independent set that spans <math>W_1+W_2</math>, it's the basis of <math>W_1+W_2</math>. We have <math>\dim (W_1+W_2)=m+n=\dim W_1+\dim W_2</math>. |
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Revision as of 18:43, 15 June 2007
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The Test
Front Page
Do not turn this page until instructed.
Math 240 Algebra I - Term Test
University of Toronto, October 24, 2006
Solve the 5 problems on the other side of this page.
Each of the problems is worth 20 points.
You have an hour and 45 minutes.
Notes.
- No outside material other than stationary and a basic calculator is allowed.
- We will have an extra hour of class time in our regular class room on Thursday, replacing the first tutorial hour.
- The final exam date was posted by the faculty - it will take place on Wednesday December 13 from 2PM until 5PM at room 3 of the Clara Benson Building, 320 Huron Street (south west of Harbord cross Huron, home of the Faculty of Physical Education and Health).
Questions Page
Solve the following 5 problems. Each of the problems is worth 20 points. You have an hour and 45 minutes.
Problem 1. Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} be a field with zero element Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0_F} , let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} be a vector space with zero element Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0_V} and let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v\in V} be some vector. Using only the axioms of fields and vector spaces, prove that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0_F\cdot v=0_V} .
Problem 2.
- In the field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb C}}
of complex numbers, compute
and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{2+3i}-\frac{1}{2-3i}} . - Working in the field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}/7} of integers modulo 7, make a table showing the values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^{-1}} for every Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a\neq 0} .
Problem 3. Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} be a vector space and let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_2} be subspaces of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} . Prove that is a subspace of iff or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_2\subset W_1} .
Problem 4. In the vector space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M_{2\times 2}({\mathbb Q})} , decide if the matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{pmatrix}1&2\\-3&4\end{pmatrix}} is a linear combination of the elements of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S=\left\{\begin{pmatrix}1&0\\-1&0\end{pmatrix},\ \begin{pmatrix}0&1\\0&1\end{pmatrix},\ \begin{pmatrix}1&1\\0&0\end{pmatrix}\right\}} .
Problem 5. Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} be a finite dimensional vector space and let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_2} be subspaces of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} for which Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_1\cap W_2=\{0\}} . Denote the linear span of by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_1+W_2} . Prove that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dim(W_1+W_2)=\dim W_1 + \dim W_2} .
The Results
Excluding a few late exams and before appeals, 63 students took the exam; The average grade was 67.46 and the standard deviation was 24.52.
The results are quite similar to what I expected them to be. The easiest questions (on average) were the computational ones, the hardest were the ones involving proofs.
How should you read your grade?
- If you got 100 you should pat yourself on your shoulder and feel good.
- If you got something like 95, you're doing great. You made a few relatively minor mistakes; find out what they are and try to avoid them next time.
- If you got something like 80 you're doing fine but you did miss something significant, probably more than just a minor thing. Figure out what it was and make a plan to fix the problem for next time.
- If you got something like 60 you should be concerned. You are still in position to improve greatly and get an excellent grade at the end, but what you missed is quite significant and you are at the risk of finding yourself far behind. You must analyze what happened - perhaps it was a minor mishap, but more likely you misunderstood something major or something major is missing in your background. Find out what it is and try to come up with a realistic strategy to overcome the difficulty!
- If you got something like 35, most likely you are not gaining much from this class and you should consider dropping it, unless you are convinced that you fully understand the cause of your difficulty (you were very sick, you really couldn't study at all for the two weeks before the exam because of some unusual circumstances, something like that) and you feel confident you have a fix for next time. If you do decide to drop the class, don't feel too bad about it. It is the hardest first year algebra class at UofT and of the thousands of students taking math here, very few come with sufficient preparation to do well in it.
Note that problems with writing are problems, period. Perhaps you got a low grade but you feel you know the material enough for a high grade only you didn't write everything you know or you didn't it write well enough or the silly graders simply didn't get what you wrote (and it isn't a simple misunderstanding - see "appeals" below). If this describes you, don't underestimate your problem. If you don't process and resolve it, it is likely to recur.
Appeals.
Remember! Grading is a difficult process and mistakes always happen - solutions get misread, parts are forgotten, grades are not added up correctly. You must read your exam and make sure that you understand how it was graded. If you disagree with anything, don't hesitate to complain! Your first stop should be the person who graded the problem in question, and only if you can't agree with him you should appeal to Dror.
Problem 1 and 2 were graded by Dmitry Donin, problem 3 was graded by Dror and problems 4 and 5 were graded by Paul Lee.
The deadline to start the appeal process is Thursday November 2 at 4PM.
Solution Set
Students are most welcome to post a solution set here.
Solutions posted by me, feel free to make any changes:
Problem 1. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0_F\cdot v=(0_F+0_F)\cdot v} (by F3)
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (0_F+0_F)\cdot v=0_F\cdot v+0_F\cdot v} (by VS8)
By VS4, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \exists\ (0_F\cdot v)' s.t. (0_F\cdot v)+(0_F\cdot v)'=0_V}
Add Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (0_F\cdot v)'} to both sides of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0_F\cdot v=0_F\cdot v+0_F\cdot v}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (0_F\cdot v)'+(0_F\cdot v)=[(0_F\cdot v)'+0_F\cdot v]+0_F\cdot v}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0_V=0_V+0_F\cdot v} (by construction)
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0_V=0_F\cdot v} (by VS3)
Problem 2. 1)
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{2+3i}-\frac{1}{2-3i}=\frac{(2-3i)-(2+3i)}{(2+3i)(2-3i)}=\frac{-6i}{2^2+3^3}=-\frac{6}{13}i}
2) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1^{-1}=1}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2^{-1}=4}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3^{-1}=5}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4^{-1}=2}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 5^{-1}=3}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 6^{-1}=6}
Problem 3. First suppose Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_1\subset W_2} or , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_1\cup W_2} equals Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_2} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_1} . Either case Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_1\cup W_2} is a subspace of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} .
Now suppose Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_1\cup W_2} is a subspace of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} and neither Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_1\subset W_2} nor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_2\subset W_1} . It must follows Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \exists\ x\in W_1} s.t. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} is not in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_2} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \exists\ y\in W_2} s.t. is not in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_1} . Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x,y\in W_1\cup W_2} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x+y\in W_1\cup W_2} . If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x+y\in W_1} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \exists\ -x\in W_1} s.t. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-x)+x+y\in W_1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-x)+x=0} , so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y\in W_1} , a contradiction. If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x+y\in W_2} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \exists\ -y\in W_2} s.t. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x+y+(-y)\in W_2} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-y)+y=0} , so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\in W_2} , a contradiction. Therefore either Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_1\subset W_2} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_2\subset W_1} .
Problem 4. Suppose Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{pmatrix}1&2\\-3&4\end{pmatrix}} is a linear combination of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{pmatrix}1&0\\-1&0\end{pmatrix}} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{pmatrix}0&1\\0&1\end{pmatrix}} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{pmatrix}1&1\\0&0\end{pmatrix}} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \exists\ a_1, a_2, a_3\in F} s.t. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{pmatrix}1&2\\-3&4\end{pmatrix}=a_1\begin{pmatrix}1&0\\-1&0\end{pmatrix}+a_2\begin{pmatrix}0&1\\0&1\end{pmatrix}+a_3\begin{pmatrix}1&1\\0&0\end{pmatrix}} .
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mbox{Need to solve}\begin{cases} 1=a_1^{}+a_3^{}\\ 2=a_2^{}+a_3^{}\\ -3=-a_1^{}\\ 4=a_2^{}\end{cases}}
Solving the equations yields Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_1=3} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_2=4} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_3=-2} , so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{pmatrix}1&2\\-3&4\end{pmatrix}} is a linear combination of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{pmatrix}1&0\\-1&0\end{pmatrix}} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{pmatrix}0&1\\0&1\end{pmatrix}} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{pmatrix}1&1\\0&0\end{pmatrix}} . Specifically, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{pmatrix}1&2\\-3&4\end{pmatrix}=3\begin{pmatrix}1&0\\-1&0\end{pmatrix}+4\begin{pmatrix}0&1\\0&1\end{pmatrix}-2\begin{pmatrix}1&1\\0&0\end{pmatrix}} .
Problem 5. Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is finite-dimensional, then so are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_2} and their basis. Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {a_1, a_2,..., a_m}} be the basis of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {b_1, b_2,..., b_n}} be the basis of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_2} so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dim W_1=m} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dim W_2=n} .
We know Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {a_1, a_2,..., a_m}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {b_1, b_2,..., b_n}} are linearly independent and clearly Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {a_1, a_2,..., a_m, b_1, b_2,..., b_n}} spans Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_1+W_2} . If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {a_1, a_2,..., a_m, b_1, b_2,..., b_n}} is linearly dependent, then there exist not all zero coefficients Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_1, c_2,..., c_{m+n}\in F} s.t. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_1 a_1+c_2 a_2+...+c_m a_m+c_{m+1} b_1+c_{m+2} b_2+...+c_{m+n} b_n=0} . Then some linear combinations of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {a_1, a_2,..., a_m}} with not all zero coefficients can be expressed in linear combinations of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {b_1, b_2,..., b_n}} , but this would imply Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_1\cap W_2\neq \{0\}} , a contradiction. Therefore is a linearly independent set that spans Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_1+W_2} , it's the basis of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_1+W_2} . We have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dim (W_1+W_2)=m+n=\dim W_1+\dim W_2} .
