## BenBlackburn 3 years ago Let V be the span of the vectors (1,3,-1,2),(-1,2,-1,-4). Express V as the solution set of a homogenous linear system.

1. colorful

I'm not quite sure I understand the question...

2. BenBlackburn

I was having the same issue. i dont even know where to start lol

3. colorful

it's asking to express the general form I suppose?$A\vec x=\vec0$and we have$A=\left[\begin{matrix}1&-1\\3&2\\-1&-1\\2&-4\end{matrix}\right]$and$\vec x=\left[\begin{array}~x_1\\x_2\\x_3\\x_4\end{array}\right]$but what does it want us to write exactly?

4. colorful

oh that matrix is sideways I think

5. colorful

no, it's just that the $$\vec x$$ is 2 rows long$A\vec x=\vec0$and we have$A=\left[\begin{matrix}1&-1\\3&2\\-1&-1\\2&-4\end{matrix}\right]$and$\vec x=\left[\begin{array}~x_1\\x_2\end{array}\right]$but what does it want us to write?

6. BenBlackburn

ummm i have no clue

7. colorful

well, the solution set is supposed to be the vector space, so we need to write the set of all solution in set notation somehow it seems

8. colorful

but it says the "homogeneous solution" so I guess let's first solve the above

9. colorful

but I'm thinking maybe homogeneous doesn't mean =0

10. BenBlackburn

ohhhh ya i think it does mean =0

11. colorful

so then I think that the span is the set$V=\{\vec x:A\vec x=\vec0\}$where$A=\left[\begin{matrix}1&-1\\3&2\\-1&-1\\2&-4\end{matrix}\right]$

12. colorful

not sure how exactly they want you to state it, but the set of all vectors that solve that system seems to be the vector space they are talking about

13. BenBlackburn

so its just saying to express the V as the solution set of a homogenious system?

14. colorful

usually the span of those vectors is the set of all linear combinations of those vectors, so you if not for the word "homogenous" I would say the span is just$S=\{c_1\langle1,3,-1,2\rangle\},c_2\langle-1,2,-1,4\rangle\}$

15. colorful

typo, I meant $S=\{c_1\langle1,3,-1,2\rangle,c_2\langle-1,2,-1,4\rangle\}$

16. colorful

hard to know what they want you to write exactly imo

17. BenBlackburn

correct i definitely see that

18. colorful

but I don't think what I just wrote is "homogenous"