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{{06-240/Navigation}} |
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===Links to Classnotes=== |
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* Classnote for Tuesday Sept 26 [http://www.megaupload.com/?d=4L41DERL] |
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* PDF file by [[User:Alla]]: [[Media:MAT_Lect005.pdf|Week 3 Lecture 1 notes]] |
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* PDF file by [[User:Gokmen]]: [[Media:06-240-Lecture-26-september.pdf|Week 3 Lecture 1 notes]] |
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---- |
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===Vector Spaces=== |
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'''Example 5.''' <br> |
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<math>\mbox{Polynomials:}{}_{}^{}</math> <br> |
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<math>7x^3+9x^2-2x+\pi\ </math> <br> |
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<math>\mbox{Let } \mathcal{F }\ \mbox{be a field.}</math> <br> |
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<math>P(\mathcal{F})=\bigg\{ \sum_{i=1}^n a_i x^i :n \in \mathbb{Z}\,\ n\ge 0\ \forall i\ \ a_i \in \mathcal{F} \bigg\} {}_{}^{} </math> |
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<math> \mbox{Addition of polynomials is defined in the expected way:}{}_{}^{} </math> <br> |
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<math> \sum_{i=0}^n a_i x^i + \sum_{i=1}^m b_i x^i =\sum_{i=0}^{max(m,n)}{(a_i+b_i)} x^i </math> <br> |
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'''Theorem 1.'''(Cancellation law for vector spaces)<br> |
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<math> \mbox{If in a vector space x+z=y+z then x=y.}{}_{}^{} </math> <br> |
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'''Proof:'''<br> |
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<math> \mbox{Add w to both sides of a given equation where w is an element}{}_{}^{} </math> |
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<math> \mbox{for which z+w=0 (exists by VS4)}{}_{}^{} </math> <br> |
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<math>(x+y)+w=(y+z)+w \ </math> <br> |
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<math> x+(z+w)=y+(z+w)\ \mbox{(by VS2)} {}_{}^{} </math> <br> |
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<math> x+0=y+0\ \mbox{(by the choice of w)} {}_{}^{} </math> <br> |
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<math> x=y\ \mbox{(by VS3)} {}_{}^{} </math> <br> |
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'''Theorem 2.''' "0 is unique" <br> |
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<math> \mbox{If some z}\in\mbox{V satisfies x+z=0 for some x}\in \mbox{V then z=0.} {}_{}^{} </math> <br> |
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'''Proof:"<br> |
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<math>x+z=x+0\ </math> <br> |
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<math>z+x=0+x\ </math> <br> |
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<math>z=0\ </math> <br> |
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'''Theorem 3.''' "negatives are unique"<br> |
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<math> \mbox{If x+y=0 and x+z=0 then y=z.} {}_{}^{} </math> <br> |
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'''Theorem 4.'''<br> |
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a)<math>0_F.x=0_V\ </math> <br> |
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b)<math>a.0_V=0_V\ </math> <br> |
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c)<math>(-a)x=a(-x)=-(ax)\ </math> <br> |
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'''Theorem 5.'''<br> |
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<math> \mbox{If } x_i\ \mbox{ i=1,...,n are in V then } \sum {x_i}=x_1+x_2+...+x_n\ \mbox{ makes sense whichever way you parse it.} {}_{}^{} </math> <br> |
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<math> \mbox{(From VS1 and VS2)} {}_{}^{} </math> <br> |
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---- |
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===Subspaces=== |
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'''Definition'''<br> |
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<math> \mbox{Let V be a vector space. A subspace of V is a subset W of V which is a vector space in itself under the operations is inherits from V.}{}_{}^{} </math> <br> |
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'''Theorem'''<br> |
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<math>W\subset V\ \mbox{is a subspace of V iff}{}_{}^{} </math> <br> |
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#<math>\forall x,y\in W\ \ x+y\in W \ </math> |
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#<math> \forall a\in F,\ \forall x\in W\ \ ax\in W\ </math> |
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#<math>0 \in W\ </math> <br> |
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'''Proof'''<br> |
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<math>\Rightarrow </math> <br> |
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<math>\mbox{Assume W is a subspace. If x,y} \in \mbox{W then x+y} \in \mbox{W because W is a vector space in itself. Likewise for a.w.}{}_{}^{} </math> <br> |
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<math>\Leftarrow </math> <br> |
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<math>\mbox{Assume W}\subset \mbox{V for which } x,y\in W\Rightarrow x+y\in W\ ; x\in W, a\in F \Rightarrow ax\in W.\ {}_{}^{} </math> <br> |
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<math>\mbox{We need to show that W is a vector space. Addition and multiplication are clearly defined on W so we just need to check VS1-VS8.}{}_{}^{} </math> <br> |
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<math> \mbox{Indeed, VS1, VS2, VS5, VS6, VS7, and VS8 hold in V hence in W.}{}_{}^{} </math> <br> |
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<math> \mbox{VS3-pick any x}\in W\ \ 0=0.x\in W\ by\ 2.\ {}_{}^{} </math> <br> |
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<math> \mbox{VS4-given x in W, take y=(-1).x}\in W\ and\ x+y=0.\ {}_{}^{} </math> <br> |
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<u>Examples</u><br> |
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'''Example 1.'''<br> |
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'''Definition'''<br> |
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<math> \mbox{If A}\in M_{m\times n}(F) \mbox{ the transpose of A, } A^t \mbox{ is the matrix } (A^t)_{ij}:=A_{ji}. {}_{}^{} </math> <br> |
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<math> \begin{pmatrix} 2 & 3 & \pi\ \\ 7 & 8 & -2 \end{pmatrix}^t = \begin{pmatrix} 2 & 7 \\ 3 & 8 \\ \pi\ & -2 \end{pmatrix} </math> <br> |
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<math> \mbox{Then:} {}_{}^{} </math> <br> |
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#<math>A^t \in M_{n\times m}(F)\ </math> <br> |
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#<math>(A^t)^t=A\ </math> <br> |
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#<math>(A+B)^t=A^t+B^t\ </math> <br> |
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#<math>(cA)^t=c(A^t)\ \forall c\in F\ </math> <br> |
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'''Definition'''<br> |
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<math>A\in M_{n\times n}(F) \mbox{ is called symmetric if } A^t=A. \ {}_{}^{} </math> <br> |
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<u>Claim</u><br> |
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<math>V=M_{n\times n}(F) \ \mbox{ is a vector space. Let } \ W=\big\{ \mbox{symmetric A-s in V}\big\} = \big\{ A\in V: A^t=A \big\}\ \mbox{ then W is a subspace of V.} {}_{}^{} </math> <br> |
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<u>Proof</u><br> |
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1.<math> \mbox{Need to show that if } A\in W and\ B\in W\ then\ A+B\in W. \ {}_{}^{} </math> <br> |
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<math>A^t=A,\ B^t=B \ </math> <br> |
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<math>(A+B)^t=A^t+B^t=A+B\ so\ A+B\in W. </math> <br> |
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<math>\mbox{If } A\in W,\ c\in F \mbox{ need to show } cA\in W {}_{}^{} </math> <br> |
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<math>(cA)^t=cA^t=cA\ \Rightarrow cA\in W </math> <br> |
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3.<math>0_M=\begin{pmatrix} 0 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & 0\end{pmatrix} \Rightarrow 0^t=0 \ so \ 0\in W</math> <br> |
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'''Example 2.'''<br> |
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<math>V=M_{n\times n}(F) </math> <br> |
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<math>A=A_{ij}\ \ trA=\sum_{i=1}^n A_{ii}\ \mbox{(the trace of A)} </math> <br> |
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<math> \mbox{Properties of tr:}{}_{}^{} </math> <br> |
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#<math>tr0_M=0 \ </math> <br> |
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#<math>tr(A+B)=tr(A)+tr(B) \ </math> <br> |
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#<math> tr(cA)=c.trA \ </math> <br> |
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<math>A=\begin{pmatrix} 1 & 0 \\ 0 & 0\end{pmatrix}\ \ B=\begin{pmatrix} 0 & 0 \\ 0 & 1\end{pmatrix} \ </math> <br> |
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<math>trA=1\ \ trB=1 \ </math> <br> |
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<math>Set\ \ W=\big\{A\in V: trA=0\big\}=\bigg\{\begin{pmatrix} 1 & 7 \\ \pi\ & -1\end{pmatrix},...\bigg\} \ </math> <br> |
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<u>Claim</u> |
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<math> \mbox{W is a subspace.}{}_{}^{} </math> <br> |
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<math> \mbox{Indeed,}{}_{}^{} </math> <br> |
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#<math>A,B\in W \Rightarrow trA=0=trB\ \ tr(A+B)=tr(A)+tr(B)=0+0=0\ so\ A+B\in W\ </math> |
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#<math>A\in W\ \ trA=0\ \ tr(cA)=c(trA)=c.0=0\ so\ cA\in W\ </math> |
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#<math>tr0_M=0\ \ 0_M\in W \ </math> |
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'''Example 3.'''<br> |
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<math> W_3=\big\{ A\in M_{n\times n}(F): trA=1\big\} \mbox{ Not a subspace.} {}_{}^{} </math> <br> |
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<math> A,B\in W_3 \Rightarrow tr(A+B)=trA+trB=1+1=2\ so\ A+B\ \not\in W_3\ </math> <br> |
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'''Theorem'''<br> |
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<math> \mbox{The intersection of two subspaces of the same space is always a subspace.}{}_{}^{}</math><br> |
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<math> \mbox{Assume }W_1\subset V \mbox{ is a subspace of V, } W_2\subset V \mbox{ is a subspace of V, then }W_1\cap W_2=\big\{ x: x\in W_1 \ and\ x\in W_2\big\} \mbox{ is a subspace.}{}_{}^{} </math> <br> |
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<math>\mbox{However, }W_1\cup W_2=\big\{x: x\in W_1\ or\ W_2\big\} \mbox{ is most often not a subspace.} {}_{}^{}</math> <br> |
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'''Proof'''<br> |
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1.<math> \mbox{Assume }x,y \in W_1\cap W_2 \mbox{ , that is, } x\in W_1, x\in W_2, y\in W_1, y\in W_2. \ {}_{}^{} </math><br> |
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<math> x+y\in W_1 \ as\ x,y\in W_1 \mbox{ and } W_1 \mbox{ is a subspace}{}_{}^{} </math><br> |
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<math> x+y\in W_2 \ as\ x,y\in W_2 \mbox{ and } W_2 \mbox{ is a subspace}{}_{}^{} </math><br> |
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<math> \mbox{So }x+y \in W_1\cap W_2. \ {}_{}^{} </math><br> |
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2.<math>\mbox{If} \ x\in W_1\cap W_2\ then\ x\in W_1 \Rightarrow cx\in W_1\ ,\ x\in W_2 \Rightarrow cx\in W_2\ \Rightarrow cx\in W_1\cap W_2. \ {}_{}^{} </math><br> |
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3.<math>0 \in W_1\ ,\ 0\in W_2 \Rightarrow 0\in W_1\cap W_2. \ </math> |
#
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Week of...
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Notes and Links
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1
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Sep 11
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About, Tue, HW1, Putnam, Thu
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2
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Sep 18
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Tue, HW2, Thu
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3
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Sep 25
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Tue, HW3, Photo, Thu
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4
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Oct 2
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Tue, HW4, Thu
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5
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Oct 9
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Tue, HW5, Thu
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6
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Oct 16
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Why?, Iso, Tue, Thu
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7
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Oct 23
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Term Test, Thu (double)
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8
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Oct 30
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Tue, HW6, Thu
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9
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Nov 6
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Tue, HW7, Thu
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10
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Nov 13
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Tue, HW8, Thu
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11
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Nov 20
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Tue, HW9, Thu
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12
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Nov 27
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Tue, HW10, Thu
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13
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Dec 4
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On the final, Tue, Thu
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F
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Dec 11
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Final: Dec 13 2-5PM at BN3, Exam Forum
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Register of Good Deeds
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![06-240-ClassPhoto.jpg](/images/thumb/8/82/06-240-ClassPhoto.jpg/180px-06-240-ClassPhoto.jpg) Add your name / see who's in!
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edit the panel
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Links to Classnotes
Vector Spaces
Example 5.
Theorem 1.(Cancellation law for vector spaces)
Proof:
Theorem 2. "0 is unique"
Proof:"
Theorem 3. "negatives are unique"
Theorem 4.
a)
b)
c)
Theorem 5.
Subspaces
Definition
Theorem
![{\displaystyle \forall x,y\in W\ \ x+y\in W\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/003928a3332df61e736d508865ce2cc04e24df1e)
![{\displaystyle \forall a\in F,\ \forall x\in W\ \ ax\in W\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2478645a03c5b1a087313fce50f4d5bba81551a)
Proof
Examples
Example 1.
Definition
Definition
Claim
Proof
1.
3.
Example 2.
Claim
![{\displaystyle A,B\in W\Rightarrow trA=0=trB\ \ tr(A+B)=tr(A)+tr(B)=0+0=0\ so\ A+B\in W\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca231973f69add6fd893af3ac3bdd00af3ee0536)
![{\displaystyle A\in W\ \ trA=0\ \ tr(cA)=c(trA)=c.0=0\ so\ cA\in W\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5190ed115228a0de3dec9bdb7ef1d8ae5bef23a5)
![{\displaystyle tr0_{M}=0\ \ 0_{M}\in W\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf315b6b8dd20ad07881ce8a1283266205d99234)
Example 3.
Theorem
![{\displaystyle {\mbox{The intersection of two subspaces of the same space is always a subspace.}}{}_{}^{}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7050ef4322d540c1fc891f4e0377b5215f48c15)
Proof
1.![{\displaystyle {\mbox{Assume }}x,y\in W_{1}\cap W_{2}{\mbox{ , that is, }}x\in W_{1},x\in W_{2},y\in W_{1},y\in W_{2}.\ {}_{}^{}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/220690af286f48da2f00130c412749fb1346b506)
![{\displaystyle x+y\in W_{1}\ as\ x,y\in W_{1}{\mbox{ and }}W_{1}{\mbox{ is a subspace}}{}_{}^{}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bece1971473484783132478844abf3a2d67071fc)
![{\displaystyle x+y\in W_{2}\ as\ x,y\in W_{2}{\mbox{ and }}W_{2}{\mbox{ is a subspace}}{}_{}^{}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0de5fb63be8a0f574ee28bd7b205bb0e754ed1c2)
![{\displaystyle {\mbox{So }}x+y\in W_{1}\cap W_{2}.\ {}_{}^{}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d87d16f54ded2a6e9d342bd22fcdb9905426fee)
2.![{\displaystyle {\mbox{If}}\ x\in W_{1}\cap W_{2}\ then\ x\in W_{1}\Rightarrow cx\in W_{1}\ ,\ x\in W_{2}\Rightarrow cx\in W_{2}\ \Rightarrow cx\in W_{1}\cap W_{2}.\ {}_{}^{}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3b8b0d40bc280a964dc90cebff325ef22f17375)
3.