09-240/Term Test - Revision history

C8sd: /* Solution Set */ Format.

2009-12-07T07:53:30Z

Solution Set: Format.

← Older revision		Revision as of 03:53, 7 December 2009
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	Therefore,		Therefore,

	<math>g(x) = 5 \cdot -\frac{1}{6}(x^3 - 3x^2 + 2x) + 4 \cdot \frac{1}{2}(x^3 - 2x^2 - x + 2) + 3 \cdot -\frac{1}{2}(x^3 - x^2 - 2x) + 8 \cdot \frac{1}{6}(x^3 - x)</math>		<math>g(x) = 5 \cdot \left( -\frac{1}{6}(x^3 - 3x^2 + 2x) \right) + 4 \cdot \frac{1}{2}(x^3 - 2x^2 - x + 2) + 3 \cdot \left( -\frac{1}{2}(x^3 - x^2 - 2x) \right) + 8 \cdot \frac{1}{6}(x^3 - x)</math>
		⚫	: <math>\,\! = x^3 - 2x + 4</math>

	Infinitely many eraser bits later...

⚫	<math>~~g(x)~~ = x^3 - 2x + 4</math>

C8sd: /* Solution Set */ Q4

2009-12-03T18:02:38Z

Solution Set: Q4

← Older revision		Revision as of 14:02, 3 December 2009
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	'''Problem 4'''		'''Problem 4'''

			Let <math>S_1 = \{ u_1, \ldots, u_n \}</math> and <math>S_2 = \operatorname{span}(S_1)</math>.

			Assume <math>S_1</math> is linearly independent. Then <math>S_1</math> is a basis, and by basis properties, <math>\forall x \in S_2, \exists</math> unique <math>a_1, \ldots, a_n \in F</math> such that <math>a_1 u_1 + \ldots + a_n u_n = x</math>. <math>\forall a \in F</math>, <math>a = 0</math> or <math>a = 1</math>, so there are two possibilities for every <math>a</math>. Therefore, <math>\|S_2\| = \underbrace{2 \times \ldots \times 2}_n = 2^n</math>, so <math>S_1</math> is linearly independent <math>\Rightarrow \|S_2\| = 2^n</math>.

			Assume <math>S_1</math> is linearly dependent. Then <math>\exists</math> non-trivial <math>a_i, \ldots, a_n</math> such that <math>a_1 u_1 + \ldots + a_n u_n = 0</math>. Since 0 has at least two representations, <math>\|S_2\| < 2^n</math>. Hence, <math>S_1</math> is linearly dependent <math>\Rightarrow \|S_2\| \ne 2^n</math>. By contrapositive, <math>\|S_2\| = 2^n \Rightarrow S_1</math> is linearly independent.

			Hence, <math>S_1</math> is linearly independent <math>\iff \|S_2\| = 2^n</math>.

	'''Problem 5'''		'''Problem 5'''

Drorbn: /* Solution Set */

2009-10-28T16:18:20Z

Solution Set

← Older revision		Revision as of 12:18, 28 October 2009
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	'''Problem 1'''		'''Problem 1'''

	If <math>a \cdot b = 0</math> and <math>a \neq 0,</math>		If <math>c \cdot v = 0</math> and <math>c \neq 0,</math>

	then <math>a^{-1} \cdot (a \cdot b) = 0;</math>		then <math>c^{-1} \cdot (c \cdot v) = 0;</math>

	thus <math>(a^{-1} \cdot a) \cdot b = 0;</math>		thus <math>(c^{-1} \cdot c) \cdot v = 0;</math>

	thus <math>1 \cdot b = 0;</math>		thus <math>1 \cdot v = 0;</math>

	thus <math>b = 0.</math>		thus <math>v = 0.</math>

	It should be remembered that "or" means "one or the other, or both."		It should be remembered that "or" means "one or the other, or both."

C8sd: /* Solution Set */ #3 copypasta

2009-10-28T02:16:39Z

Solution Set: #3 copypasta

← Older revision		Revision as of 22:16, 27 October 2009
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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>.
	'''Problem 3'''



			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 follow that <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>.

	'''Problem 4'''		'''Problem 4'''

Sknad: /* Solution Set */

2009-10-27T19:59:53Z

Solution Set

← Older revision		Revision as of 15:59, 27 October 2009
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	It should be remembered that "or" means "one or the other, or both."		It should be remembered that "or" means "one or the other, or both."


			'''Problem 2'''

			(1)

			<math>4i(1 + i) = 4i + 4i^2 = -4 + 4i</math>

			<math>\frac{4i}{1+i} = \frac{4i(1 - i)}{(1 + i)(1 - i)} = \frac{4 + 4i}{2} = 2 + 2i</math>


			(2)

			If <math>z^2 = 2i</math> then <math>z = 1 + i \Rightarrow (1 + i)^2 = 1 + 2i + i^2 = 2i</math>


			(3)

			<math>x = 8, 3</math>


			'''Problem 3'''




			'''Problem 4'''



			'''Problem 5'''

			Recall that<math> f_i(x) = \prod_{k = 0 k \neq i}^{n}{\frac{x - c_k}{c_i - c_k}}</math> and that <math>g(x) = \sum_{i = 0}^{n}{b_if_i}</math>

			After tedious computation with the addition of eraser bits covering your paper, you should obtain the following:

			<math>f_0(x) = -\frac{1}{6}(x^3 - 3x^2 + 2x)</math>

			<math>f_1(x) = \frac{1}{2}(x^3 - 2x^2 - x + 2)</math>

			<math>f_2(x) = -\frac{1}{2}(x^3 - x^2 - 2x)</math>

			<math>f_3(x) = \frac{1}{6}(x^3 - x)</math>

			Therefore,

			<math>g(x) = 5 \cdot -\frac{1}{6}(x^3 - 3x^2 + 2x) + 4 \cdot \frac{1}{2}(x^3 - 2x^2 - x + 2) + 3 \cdot -\frac{1}{2}(x^3 - x^2 - 2x) + 8 \cdot \frac{1}{6}(x^3 - x)</math>

			Infinitely many eraser bits later...

			<math>g(x) = x^3 - 2x + 4</math>

Sknad: /* Solution Set */

2009-10-27T19:24:55Z

Solution Set

← Older revision		Revision as of 15:24, 27 October 2009
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	==Solution Set==		==Solution Set==
	Students are most welcome to post a solution set here.		Students are most welcome to post a solution set here.


			'''Problem 1'''

			If <math>a \cdot b = 0</math> and <math>a \neq 0,</math>

			then <math>a^{-1} \cdot (a \cdot b) = 0;</math>

			thus <math>(a^{-1} \cdot a) \cdot b = 0;</math>

			thus <math>1 \cdot b = 0;</math>

			thus <math>b = 0.</math>

			It should be remembered that "or" means "one or the other, or both."

Drorbn at 16:33, 27 October 2009

2009-10-27T16:33:10Z

← Older revision		Revision as of 12:33, 27 October 2009
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	In style and spirit this exam will not be very different of the one I gave 3 years ago. See [[06-240/Term Test]].		In style and spirit this exam will not be very different of the one I gave 3 years ago. See [[06-240/Term Test]].

	{{06-240/Navigation}}

	==The Test==		==The Test==

Drorbn at 16:23, 27 October 2009

2009-10-27T16:23:26Z

← Older revision		Revision as of 12:23, 27 October 2009
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	In style and spirit this exam will not be very different of the one I gave 3 years ago. See [[06-240/Term Test]].		In style and spirit this exam will not be very different of the one I gave 3 years ago. See [[06-240/Term Test]].

			{{06-240/Navigation}}

			==The Test==

			===Front Page===

			<center>

			'''Do not turn this page until instructed.'''

			<font style="font-size:150%">Math 240 Algebra I - Term Test</font>


			University of Toronto, October 22, 2009

			'''Solve the 5 problems on the other side of this page.'''

			Each of the problems is worth 20 points.

			You have an hour and 50 minutes.

			'''Notes.'''

			</center>

			* No outside material other than stationary and a basic calculator is allowed.
			* We will have an hour of discussion time right after this test.
			* The final exam date was posted by the faculty --- it will take place on Wednesday December 16 from 9AM until noon at room BN2S of the Clara Benson Building, 320 Huron Street (south west of Harbord cross Huron, home of the Faculty of Physical Education and Health).

			<center> '''Good Luck!''' </center>

			===Questions Page===

			'''Solve the following 5 problems.''' Each of the problems is worth 20 points. You have an hour and 50 minutes.

			'''Problem 1.''' Let <math>V</math> be a vector space over a field <math>F</math>, let <math>c\in F</math> and let <math>v\in V</math>. Prove that if <math>cv=0</math>, then either <math>c=0</math> or <math>v=0</math>.

			'''Problem 2.'''
			# In the field <math>{\mathbb C}</math> of complex numbers, compute
			<center><math>4i(1+i)</math>      and     <math>\frac{4i}{1+i}</math>.</center>
			(To be precise, "compute" means "write in the form <math>a+ib</math>, where
			<math>a,b\in{\mathbb R}</math>").
			# In the field <math>{\mathbb C}</math> of complex numbers, find an element <math>z</math> so that <math>z^2=2i</math>.
			# In the 11-element field <math>F_{11}</math> of remainders modulo 11, find
			all solutions of the equation <math>x^2=-2</math>.

			'''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>.

			'''Problem 4.''' Let <math>u_1,\ldots,u_n</math> be vectors in a
			vector space over the field with two element <math>F_2</math>. Show that the number of
			elements in the set <math>\mbox{Span}\{u_1,\dots,u_n\}</math> is equal to <math>2^n</math> if and only
			if <math>u_1,\ldots,u_n</math> are linearly independent..

			'''Problem 5.''' Find a polynomial <math>f\in P_3({\mathbb R})</math> that
			satisfies <math>f(-1)=5</math>, <math>f(0)=4</math>, <math>f(1)=3</math>, and <math>f(2)=8</math>.

			<center> '''Good Luck!''' </center>

			{{09-240/Results of the Term Test}}

			==Solution Set==
			Students are most welcome to post a solution set here.

Drorbn: /* Announcement */

2009-10-15T03:27:27Z

Announcement

← Older revision		Revision as of 23:27, 14 October 2009
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	==Announcement==		==Announcement==

	Our one and only Term Test is coming up. It will take place in class on Thursday October 21 2009, starting promptly at 1:10PM and ending at 3:00PM sharp, in our normal classroom, MP103. It will consist of 4-5 questions (each may have several parts) on everything that will be covered in class by October 16: the axiomatic definition of fields and some basic properties of fields, <math>{\mathbb C}</math> and other examples, a tiny bit on the field with <math>p</math> elements <math>F_p</math>, the axiomatic definition of vector spaces, basic properties and examples of vector spaces, spans, linear combinations and linear equations, linear dependence and independence, bases, the replacement lemma and its consequences, a bit about linear transformations and a few smaller topics that we touched but that do not deserve their own headers.		Our one and only Term Test is coming up. It will take place in class on Thursday October 22 2009, starting promptly at 1:10PM and ending at 3:00PM sharp, in our normal classroom, MP103. It will consist of 4-5 questions (each may have several parts) on everything that will be covered in class by October 16: the axiomatic definition of fields and some basic properties of fields, <math>{\mathbb C}</math> and other examples, a tiny bit on the field with <math>p</math> elements <math>F_p</math>, the axiomatic definition of vector spaces, basic properties and examples of vector spaces, spans, linear combinations and linear equations, linear dependence and independence, bases, the replacement lemma and its consequences, a bit about linear transformations and a few smaller topics that we touched but that do not deserve their own headers.

	Note that there may be some computations, but nothing that will require a calculator. Note also that I may include some questions from the homework assignments verbatim or nearly verbatim.		Note that there may be some computations, but nothing that will require a calculator. Note also that I may include some questions from the homework assignments verbatim or nearly verbatim.

Drorbn at 23:45, 14 October 2009

2009-10-14T23:45:37Z

New page

{{09-240/Navigation}}

==Announcement==

Our one and only Term Test is coming up. It will take place in class on Thursday October 21 2009, starting promptly at 1:10PM and ending at 3:00PM sharp, in our normal classroom, MP103. It will consist of 4-5 questions (each may have several parts) on everything that will be covered in class by October 16: the axiomatic definition of fields and some basic properties of fields, <math>{\mathbb C}</math> and other examples, a tiny bit on the field with <math>p</math> elements <math>F_p</math>, the axiomatic definition of vector spaces, basic properties and examples of vector spaces, spans, linear combinations and linear equations, linear dependence and independence, bases, the replacement lemma and its consequences, a bit about linear transformations and a few smaller topics that we touched but that do not deserve their own headers.

Note that there may be some computations, but nothing that will require a calculator. Note also that I may include some questions from the homework assignments verbatim or nearly verbatim.

;Will there be "proof questions"?
:Sure. What else have we done so far?
;Do we need to know the proofs from class?
:Sure. There's a reason why these proofs are in class to start with; if they weren't valuable, we wouldn't have covered them.

No electronic devices capable of displaying text or sounding speech will be allowed.

In style and spirit this exam will not be very different of the one I gave 3 years ago. See [[06-240/Term Test]].