C8sd: /* Linear combinations */ Formatting.

2009-12-07T05:29:42Z

Linear combinations: Formatting.

← Older revision		Revision as of 01:29, 7 December 2009
Line 47:		Line 47:
	: <math>u = a_1 u_1 + a_2 u_2 + \ldots + a_n u_n</math>		: <math>u = a_1 u_1 + a_2 u_2 + \ldots + a_n u_n</math>

	'''Example''': <math>\mathbb P_n(F) = \{ \mbox{Polynomials of degree at most } n \mbox{ with coefficients in } ''F'' \}</math><br />		'''Example''': <math>\mathbb P_n(F) = \{ \mbox{Polynomials of degree at most } n \mbox{ with coefficients in } F \}</math><br />
	<math>= \{ \sum_{i=0}^n a_i x^i : a_i \in F \}</math>		<math>= \left\{ \sum_{i=0}^n a_i x^i : a_i \in F \right\}</math>

	'''Definition''': A subset <math>S \subset \mathbf V</math> "generates" or "spans" '''V''' iff the set of linear combinations of elements of ''S'' is all of ''V''.		'''Definition''': A subset <math>S \subset \mathbf V</math> "generates" or "spans" '''V''' iff the set of linear combinations of elements of ''S'' is all of '''V'''.

	'''Example''': Let <math>\mathbf V = \mathbb M_{n \times n}(\real)</math><br />		'''Example''': Let <math>\mathbf V = \mathbb M_{n \times n}(\real)</math><br />
Line 56:		Line 56:
	Then <math>S = \{ M_1, M_2, M_3, M_4 \}</math> generates '''V'''.		Then <math>S = \{ M_1, M_2, M_3, M_4 \}</math> generates '''V'''.

	Proof: Given <math>\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathbb M_{2 \times 2}(\real)~~, write~~</math> <br />		Proof: Given <math>\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathbb M_{2 \times 2}(\real)</math>, write<br />
	<math>\begin{pmatrix} a & b \\ c & d \end{pmatrix} = aM_1 + bM_2 + cM_3 + dM_4.</math>		<math>\begin{pmatrix} a & b \\ c & d \end{pmatrix} = aM_1 + bM_2 + cM_3 + dM_4.</math>

C8sd: Add partial notes.

2009-12-07T05:28:27Z

Add partial notes.

← Older revision		Revision as of 01:28, 7 December 2009
Line 10:		Line 10:
	Image:September 29 2009 Lecture Notes Page 6.jpg\|Page 6		Image:September 29 2009 Lecture Notes Page 6.jpg\|Page 6
	</gallery>		</gallery>

			== Vector subspaces ==

			'''Definition'''. <math>\mathbf W \subset \mathbf V</math> is a "subspace" if it is a vector space under the operations it inherits from '''V'''.

			'''Theorem'''. <math>\mathbf W \subset \mathbf V</math> is a subspace iff it is "closed under addition and vector multiplication by scalars", i.e. <math>x, y \in \mathbf W \Rightarrow x + y \in \mathbf W</math> and <math>a \in F, x \in \mathbf W \Rightarrow ax \in \mathbf W</math>.

			'''Goal''': Every VS has a "basis", so while we don't ''have'' to use coordinates, we always can.

			'''Examples''' of what is not a subspace (without diagrams):
			# A unit circle is not closed under addition of scalar multiplication.
			# The x-axis <math>\cup</math> y-axis is closed under scalar multiplication, but not under addition.
			# A single quadrant of the Cartesian plane is closed under addition, but not under scalar multiplication.

			'''Examples''' of subspaces:
			# <math>\{0\}</math>
			# Any VS (which is a subspace of itself)
			# A line passing through the origin (if it does not pass through the origin, then it is not closed under scalar multiplication)
			# A plane
			# Let <math>\mathbf V = \mathbb M_{n \times n}(F)</math>. If <math>W = \{ A \in \mathbf V : A^\top = A \}</math>, then '''W''' is a subspace of '''V'''. ('''W''' is the set of "symmetric" matrices in '''V'''; ''A''<sup>T</sup> denotes the [http://en.wikipedia.org/wiki/Transpose transpose] of ''A''.)
			# <math>\mathbf W = \{ A \in \mathbb M_{n \times n} : \operatorname{tr}(A) = 0 \}</math>
			#: where <math>\operatorname{tr}(A) = \sum_{i=1}^n a_{ii}</math> is the "trace" of ''A''.
			#: Properties of trace:
			## <math>\operatorname{tr}(0 \cdot A) = 0</math>
			## <math>\operatorname{tr}(cA) = c \cdot \operatorname{tr}(A)</math>
			## <math>\operatorname{tr}(A + B) = \operatorname{tr}(A) + \operatorname{tr}(B)</math>
			#: so '''W''' is indeed a subspace.

			'''Claim''': If '''W<sub>1</sub>''' and '''W<sub>2</sub>''' are subspaces of '''V''', then
			# <math>W_1 \cap W_2 = \{ x \in \mathbf V : x \in \mathbf W_1 \mbox{ and } \mathbf W_2 \}</math> is a subspace of '''V''', '''W<sub>1</sub>''', and '''W<sub>2</sub>'''.
			# But <math>W_1 \cup W_2 = \{ x \in \mathbf V : x \in \mathbf W_1 \mbox{ or } x \in \mathbf W_2 \}</math> is a subspace of '''V''' iff <math>\mathbf W_1 \subset \mathbf W_2</math> or <math>\mathbf W_2 \subset \mathbf W_1</math>. (See [[09-240:HW2\|HW2]] pp. 20-21, #19.)

			== Linear combinations ==

			'''Definition''': A vector ''u'' is a "linear combination" (l.c.) of vectors ''u''<sub>1</sub>, ..., ''u''<sub>n</sub> if there exists scalars ''a''<sub>1</sub>, ..., ''a''<sub>n</sub> such that
			: <math>u = a_1 u_1 + a_2 u_2 + \ldots + a_n u_n</math>

			'''Example''': <math>\mathbb P_n(F) = \{ \mbox{Polynomials of degree at most } n \mbox{ with coefficients in } ''F'' \}</math><br />
			<math>= \{ \sum_{i=0}^n a_i x^i : a_i \in F \}</math>

			'''Definition''': A subset <math>S \subset \mathbf V</math> "generates" or "spans" '''V''' iff the set of linear combinations of elements of ''S'' is all of ''V''.

			'''Example''': Let <math>\mathbf V = \mathbb M_{n \times n}(\real)</math><br />
			Let <math>M_1 = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, M_2 = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, M_3 = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}, M_4 = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}</math><br />
			Then <math>S = \{ M_1, M_2, M_3, M_4 \}</math> generates '''V'''.

			Proof: Given <math>\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathbb M_{2 \times 2}(\real), write</math> <br />
			<math>\begin{pmatrix} a & b \\ c & d \end{pmatrix} = aM_1 + bM_2 + cM_3 + dM_4.</math>

			'''Example''': Let <math>N_1 = \begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix}, N_2 = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}, N_3 = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}, N_4 = \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}</math><br />
			Does <math>\{ N_1, N_2, N_3, N_4 \}</math> generate '''V'''? <br />
			: <math>M_1 = -\frac23 N_1 + \frac13(N_2 + N_3 + N_4)</math>
			: <math>M_2 = -\frac23 N_2 + \frac13(N_1 + N_3 + N_4)</math>
			: <math>M_3 = -\frac23 N_3 + \frac13(N_1 + N_2 + N_4)</math>
			: <math>M_4 = -\frac23 N_4 + \frac13(N_1 + N_2 + N_3)</math>
			Then <math>\begin{pmatrix} a & b \\ c & d \end{pmatrix} = aM_1 + bM_2 + cM_3 + dM_4 =</math>
			: <math>a \cdot \left( -\frac23 N_1 + \frac13(N_2 + N_3 + N_4) \right) + b \cdot \left( -\frac23 N_2 + \frac13(N_1 + N_3 + N_4) \right) + \ldots</math>

			'''Theorem''': If <math>S \in \mathbf V</math>, then <math>\operatorname{span}(S) = </math> {all l.c. of elements of ''S''} is a subspace of '''V'''.

C8sd: Yangjiay's notes.

2009-10-02T22:21:33Z

Yangjiay's notes.

New page

{{09-240/Navigation}}
{{09-240/Class Notes Warning}}

<gallery>
Image:September 29 2009 Lecture Notes Page 1.jpg|[[User:Yangjiay|Yangjiay]] - Page 1
Image:September 29 2009 Lecture Notes Page 2.jpg|Page 2
Image:September 29 2009 Lecture Notes Page 3.jpg|Page 3
Image:September 29 2009 Lecture Notes Page 4.jpg|Page 4
Image:September 29 2009 Lecture Notes Page 5.jpg|Page 5
Image:September 29 2009 Lecture Notes Page 6.jpg|Page 6
</gallery>

09-240/Classnotes for Tuesday September 29 - Revision history

C8sd: /* Linear combinations */ Formatting.

C8sd: Add partial notes.

C8sd: Yangjiay's notes.