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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.
![{\displaystyle {\mbox{There is a special force vector called 0.}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b179684b4e3625c2b17452dd22eed60126ee64ee)
![{\displaystyle {\mbox{They can be added.}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/12a84f78684fdcdf738c1a802f773b2a2d0c5fc3)
![{\displaystyle {\mbox{They can be multiplied by any scalar.}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a590eca541009903319040562be774bb122dc30c)
Properties
![{\displaystyle xy=yx\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffb66f15ee161fd191d795cfc2d58c862a9f1a8e)
![{\displaystyle x(yz)=(xy)z\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3eb552bc8aeadd50a13fb3e46b720f2f2a6f9b6a)
![{\displaystyle x0=x\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d38bda03b6102172f86d844da505111776b0534)
![{\displaystyle \forall x\;\exists \ y\ {\mbox{ s.t. }}xy=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b55179cec5e99f4f7cdce22082fd67ccb102276)
![{\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(xy)=axay\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/0bdc08c988efb650528abeae4e7518e7109de1b8)
![{\displaystyle (ab)x=axbx\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6302f67903264595722a0046d2cd7dc775934df)
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/0534c02fd9895b2afcd2449fa735c697324bcbce)
![{\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\ xy=yx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae959bc01644b22d4208e1ad96ce3bbca0f93f99)
![{\displaystyle \forall x,y\in V\ x(yz)=(xy)z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20d35356225da86a25e2c5a8f8e002e61a0c70c6)
![{\displaystyle \exists \ 0\in Vs.t.\;\forall x\in V\ x0=x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c9d1361c884971b9dfcff2200605b6dc86531db)
![{\displaystyle \forall x\in V\;\exists \ y\in V\ s.t.\ xy=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/adb966eaedd204a8c22eaf47aef4b02217f04333)
![{\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(xy)=axay\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/0bdc08c988efb650528abeae4e7518e7109de1b8)
![{\displaystyle \forall x\in V\ ,\forall a,b\in F\ (ab)x=axbx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b362c5aec5b66b66ff002c4c31d8c086bb7443d)
9.
Examples
Ex.1.
Ex.2.
Failed to parse (unknown function "\begin{pmatrix}"): {\displaystyle V=M_{m\times n}(F)=\left\lbrace\begin{pmatrix} a_{11}}