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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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Scan of Lecture Notes
Scan of Tutorial notes
Force Vectors
A force has a direction and a magnitude.
- There is a special force vector called 0.
- They can be added.
- They can be multiplied by any scalar.
Properties
(convention:
are vectors;
are scalars)
![{\displaystyle x+y=y+x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/34531a1a1d79d62f63926487d85bcd05ed2bb3ab)
![{\displaystyle x+(y+z)=(x+y)+z\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf050bca647083605a3f536711f938c78f279146)
![{\displaystyle x+0=x\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/40c69b5b348f9ca362a4e695ff90059ba9402de9)
![{\displaystyle \forall x\;\exists \ y\ {\mbox{ s.t. }}x+y=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6eb663fdb476ddb23a8fc81e33ff1eaaad8f749f)
![{\displaystyle 1\cdot x=x\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a95288ff50bbcf579dffc062844238807d42654f)
![{\displaystyle a(bx)=(ab)x\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7523325fb2d65ffe394a7c931f85c2855765c4b3)
![{\displaystyle a(x+y)=ax+ay\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/db5d5a003933d5b90bfe76e3e2cb292316cc3452)
![{\displaystyle (a+b)x=ax+bx\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b9a06e8cbd0b15dddc24c6d3343d9ead4c136f4b)
Definition
Let F be a field "of scalars". A vector space over F is a set V, of "vectors", along with two operations
![{\displaystyle +:V\times V\to V}](https://wikimedia.org/api/rest_v1/media/math/render/svg/982bd4f2bd4daba2cb683b2ea4ac907d5422dcce)
![{\displaystyle \cdot :F\times V\to V{\mbox{, so that:}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d17090c22bc1ea51ea36ba24035a9785252add5)
![{\displaystyle \forall x,y\in V\ x+y=y+x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ebfcbab8f76165500ffe744313298026091e978)
![{\displaystyle \forall x,y\in V\ x+(y+z)=(x+y)+z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ddd1a9bbeb7293e94c72b05c3cc122b9f4652b87)
![{\displaystyle \exists \ 0\in Vs.t.\;\forall x\in V\ x+0=x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0f62a3124291a3c4614dd490089b7822baa8bb5)
![{\displaystyle \forall x\in V\;\exists \ y\in V\ s.t.\ x+y=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be891d4519d5044e7619c7948eb77cb7380135fc)
![{\displaystyle 1\cdot x=x\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a95288ff50bbcf579dffc062844238807d42654f)
![{\displaystyle a(bx)=(ab)x\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7523325fb2d65ffe394a7c931f85c2855765c4b3)
![{\displaystyle a(x+y)=ax+ay\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/db5d5a003933d5b90bfe76e3e2cb292316cc3452)
![{\displaystyle \forall x\in V\ ,\forall a,b\in F\ (a+b)x=ax+bx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65d86e129b5d7dfc56c0e12e61e2001f04dff51f)
9.
Examples
Ex.1.
Ex.2.
Ex.3.
form a vector space over
.
Ex.4.
Ex.5.
is a vector space over
.
Ex.6.