1. ## Span (Lin Alg)

Given the following vectors:

v_1 = [1, -2, 3]

v_2 = [-1, 1, 0]

v_3 = [1, -3, 5]

Find a vector w in R^3 (which is not a scalar multiple of v_1, v_2, or v_3) that's in the Span{v_1, v_2, v_3} but not in the Span{v_1, v_2}.

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Not sure. Is it asking me to find an augmented vector so when I row reduce the 3x4 matrix it's consistent, but when I take that same vector and augment it with v_1, v_2 (so I'd have a 2x4 matrix) it's not consistent? Not sure...

2. Can anyone point me in the right direction?

3. Originally Posted by Ideasman
Given the following vectors:

v_1 = [1, -2, 3]

v_2 = [-1, 1, 0]

v_3 = [1, -3, 5]

Find a vector w in R^3 (which is not a scalar multiple of v_1, v_2, or v_3) that's in the Span{v_1, v_2, v_3} but not in the Span{v_1, v_2}.
I haven't bothered to show it but visually it looks like your 3 vectors span $\displaystyle \mathbb{R} ^3$, so essentially the question is asking for a vector in $\displaystyle \mathbb{R} ^3$ that is not in the plane formed by $\displaystyle \text{Span} \{ v_1, v_2 \}$. ie. find a vector perpendicular to the plane formed by $\displaystyle v_1, v_2$.

(Of course, you need to write this vector in terms of your basis.)

-Dan

4. Originally Posted by topsquark
I haven't bothered to show it but visually it looks like your 3 vectors span $\displaystyle \mathbb{R} ^3$, so essentially the question is asking for a vector in $\displaystyle \mathbb{R} ^3$ that is not in the plane formed by $\displaystyle \text{Span} \{ v_1, v_2 \}$. ie. find a vector perpendicular to the plane formed by $\displaystyle v_1, v_2$.

(Of course, you need to write this vector in terms of your basis.)

-Dan
I'm going crazy!

I confirmed that v_1, v_2, v_3 do span R^3.

Wouldn't ANY vector work, mean we have 2 vectors, and if we AUGMENT it with a 3rd, it will ALWAYS be inconsistent...but if we augment it with the matrix we already know spans R^3, wouldn't ANY vector work for my question...grrr confusing.
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