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'''Definition.''' <math> v\in V</math>is a linear combination of elements in <math>S\ subset V</math> if <math>\exists u_1,\ldots,u_n\in S </math> and <math>a_1,\dots,a_n \in F </math> such that <math>V=\sum a_i u_i</math> |
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<math>\mbox{Definition}{}_{}^{}</math> |
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<math>v\in V \mbox{ is a linear combination of elements in } S\subset V</math> |
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'''Example.''' In <math>P_3(\mathbb{R})</math>, <math>v_1=2x^3-2x^2+12-6</math> is a linear combination of: <math>u_1=x^3-2x^2-5x-3</math> and <math>u_2=3x^3-5x^2-4x-9</math> but <math>v_2=3x^3-2x^2+7x+8</math> is not. |
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<math> \ mbox{ if }\exists u_1,\ldots,u_n\in S \mbox{ and } a_1,\dots,a_n \in F \mbox{ such that } V=\sum a_i u_i</math> |
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<math>\mbox{Example}{}_{}^{}</math> |
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'''Why?''' <math>v_1=2x^3-2x^2+12-6=a_1u_1+a_2u_2</math> |
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<math>= a_1(x^3-2x^2-5x-3 )+a_2(3x^3-5x^2-4x-9 )</math> |
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<math>\mbox{In }P_3(\mathbb{R})\mbox{,}</math> |
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<math>v_1^{}=2x^3-2x^2+12-6 \mbox{ is a linear combination of:}</math> |
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<math> u_1^{}=x^3-2x^2-5x-3 \mbox{ and }u_2=3x^3-5x^2-4x-9</math> |
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<math>\mbox{but } v_2^{}=3x^3-2x^2+7x+8 \mbox{ is not.}</math> |
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<math>\mbox{Why?}{}_{}^{}</math> |
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<math>v_1^{}=2x^3-2x^2+12-6=a_1^{}u_1+a_2u_2=a_1(x^3-2x^2-5x-3)+a_2(3x^3-5x^2-4x-9)</math> |
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<math>v_1^{}=-4u_1+2u_2</math> |
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<math>v_1^{}=-4u_1+2u_2</math> |
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<math>\mbox{Definition}{}_{}^{}</math> |
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<math>\mbox{Definition}{}_{}^{}</math> |
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<math>\mbox{We say that a subset }S\subset V\mbox{ generates or spans }V \mbox{ if span }S=\lbrace\mbox{ all linear combinations of elements in } S\rbrace=V</math> |
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<math>\mbox{We say that a subset }S\subset V\mbox{ generates or spans }V </math> |
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<math>\mbox{ if span }S=\lbrace\mbox{ all linear combinations of elements in } S\rbrace=V{}_{}^{}</math> |
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<math>\mbox{Examples}{}_{}^{}</math> |
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<math>\mbox{Examples}{}_{}^{}</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
Definition.
is a linear combination of elements in
if
and
such that
Example. In
,
is a linear combination of:
and
but
is not.
Why?
![{\displaystyle \lbrace M_{1}^{},M_{2},M_{3},M_{4}\rbrace {\mbox{ generates }}V}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d9ee66b37301b0c70995d4b927db9bc38813a884)
![{\displaystyle \lbrace N_{1}^{},N_{2},N_{3},N_{4}\rbrace {\mbox{ generates }}V}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd2998a8ab879a6741890ba6e258cf0cade7774f)
![{\displaystyle \lbrace M_{1}^{},M_{2},M_{3}\rbrace {\mbox{ does not generate }}V}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1df7a2f77e52b2476302198d47906618831907bd)
![{\displaystyle \lbrace N_{1}^{},N_{2},N_{3}\rbrace {\mbox{ does not generate }}V}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a33fd3560889e7525c1e9cb81e2cc859acdb6319)
![{\displaystyle \emptyset \subset V{\mbox{ is linearly independent}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2eaa82f8191774ed56c5b9190e041bf20b88a3a)
![{\displaystyle {\mbox{Suppose }}u\in V,\quad \lbrace u\rbrace {\mbox{ the singleton set is linearly independent iff }}u_{}^{}\neq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6817fb0fe044e0a5dcf757157d1cd47f724b9505)