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Name two pairs of vectors that could span `RR^2`.  Show how the vector `(3, 2)` could...

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user3666642 | (Level 1) eNoter

Posted June 10, 2013 at 9:07 PM via web

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Name two pairs of vectors that could span `RR^2`.  Show how the vector `(3, 2)` could be written as a linear combination of your spanning set.

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degeneratecircle | High School Teacher | (Level 2) Associate Educator

Posted June 10, 2013 at 11:37 PM (Answer #1)

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One pair is the standard basis that consists of the vectors  `e_1=(1,0)` and `e_2=(0,1).` In this basis, `(3,2)=3e_1+2e_2.` 

Another pair that spans `RR^2` is `b_1=(3,2)` and `b_2=(1,1).` They do indeed span `RR^2` because neither is a multiple of the other. In this basis, `(3,2)=1b_1+0b_2.` 

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