## BenBlackburn Group Title 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. one year ago one year ago

1. colorful Group Title

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

2. BenBlackburn Group Title

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

3. colorful Group Title

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 Group Title

oh that matrix is sideways I think

5. colorful Group Title

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 Group Title

ummm i have no clue

7. colorful Group Title

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 Group Title

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

9. colorful Group Title

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

10. BenBlackburn Group Title

ohhhh ya i think it does mean =0

11. colorful Group Title

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 Group Title

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 Group Title

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

14. colorful Group Title

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 Group Title

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

16. colorful Group Title

hard to know what they want you to write exactly imo

17. BenBlackburn Group Title

correct i definitely see that

18. colorful Group Title

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