Difference between revisions of "06-240/Classnotes For Tuesday October 3"
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===Links to Classnotes=== | ===Links to Classnotes=== | ||
* PDF file by [[User:Alla]]: [[Media:MAT_Lect007.pdf|Week 4 Lecture 1 notes]] | * PDF file by [[User:Alla]]: [[Media:MAT_Lect007.pdf|Week 4 Lecture 1 notes]] | ||
+ | * PDF file by [[User:Gokmen]]: [[Media:06-240-Lecture-03-October.pdf|Week 4 Lecture 1 notes]] | ||
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− | <math>\ | + | '''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> |
− | <math> | + | '''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. |
− | <math> | + | |
− | <math> | + | '''Why?''' <math>v_1=2x^3-2x^2+12-6=a_1u_1+a_2u_2</math> |
+ | <math>=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> | <math>v_1^{}=-4u_1+2u_2</math> | ||
<math>\mbox{Definition}{}_{}^{}</math> | <math>\mbox{Definition}{}_{}^{}</math> | ||
− | <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> | + | <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> | ||
<math>\mbox{Examples}{}_{}^{}</math> | <math>\mbox{Examples}{}_{}^{}</math> |
Latest revision as of 08:14, 12 July 2007
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Links to Classnotes
- PDF file by User:Alla: Week 4 Lecture 1 notes
- PDF file by User:Gokmen: Week 4 Lecture 1 notes
Definition. is a linear combination of elements in if and such that
Example. In , is a linear combination of: and but is not.
Why?