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Show that the span of the vectors `<<1,4,9>>` , `<<1,2,3>>` and...

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gomezzzzzzzzz... | Student | eNoter

Posted March 15, 2013 at 11:45 PM via web

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Show that the span of the vectors `<<1,4,9>>` , `<<1,2,3>>` and `<<1,1,1>>` is all of `RR^3` .

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tiburtius | High School Teacher | (Level 3) Associate Educator

Posted March 16, 2013 at 7:02 AM (Answer #1)

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You have 3 vectors from `RR^3` if all 3 vectors are linearly independent then they make a base for `RR^3` that is they span `RR^3`.

Vectors `v_1,v_2,ldots,v_n` are linearly independent if

`a_1 v_1+a_2 v_2+cdots+a_n v_n=0`

implies that

`a_1=a_2=ldots=a_n=0`

Let's check if our vectors are linearly independent.

`a<<1,4,9>>+b<<1,2,3>>+c<<1,1,1>>=0`

From this we get system of 3 equations (one for each coordinate) with 3 variables.

`a+b+c=0`

`4a+2b+c=0`

`9a+3b+c=0`

Now you can solve this system of equations by using Gauss elimination. 

System has unique solution `a=b=c=0` which means that all three vectors are independent and thus they span `RR^3`  ` ` which is what we were supposed to prove.

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