# 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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Expert Answers

tiburtius | Certified Educator

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.**