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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.