Determining if a vector belongs in a subspace

[burt]

I was given the above question. I am not sure how to solve it. How do I determine if a vector is in the subspace generated by other vectors?

 

[yoscar04]

You need to prove that u can be written as a linear combination of v₁, v₂ and v₃. Do you know what a linear combination is? Try to do what I just wrote you. If you get stuck, we will try to help you.

 

[HallsofIvy]

In other words, do there exist numbers, a, b, and c, such that
a+2b+ 7c= -1
-2a+ 5b+ 19c= -13
3a+ 4b+ 17c= -3
2a- 2c= 6

Solve any three of those equations then see if those values of a, b, and c make the remaining equation true.

 

[burt]

In other words, do there exist numbers, a, b, and c, such that
a+2b+ 7c= -1
-2a+ 5b+ 19c= -13
3a+ 4b+ 17c= -3
2a- 2c= 6

Solve any three of those equations then see if those values of a, b, and c make the remaining equation true.

So can I just set it up as a matrix with u as my solution set?

 

[burt]

I set up a matrix and ended up with my bottom row showing 0=0. Does this make a difference if I can still find a solution for every x? At first I thought it would mean the vector does not fit, but now I see it just means that not all vectors will fit - but this one still can.

 

[yoscar04]

I set up a matrix and ended up with my bottom row showing 0=0. Does this make a difference if I can still find a solution for every x? At first I thought it would mean the vector does not fit, but now I see it just means that not all vectors will fit - but this one still can.

You are supposed to find the values of a, b and c. If there are no values that fulfill the system of equations then the vector u doesn't belong to the subspace generated by the v_i's. You can try to solve even by simple substitution. What x are you talking about?

 

[burt]

You are supposed to find the values of a, b and c. If there are no values that fulfill the system of equations then the vector u doesn't belong to the subspace generated by the v_i's. You can try to solve even by simple substitution. What x are you talking about?

I set up a matrix:
[MATH]\left[\begin{matrix}1&2&7&-1\\ 2(-1)&5&19&-13\\3&4&17&-3\\2&0&-2&6\\\end{matrix}\right][/MATH]In REF this becomes:
\(\left[\begin{matrix}1&\frac43&\frac{17}{3}&-1\\0&1&\frac{91}{23}&\frac{-45}{23}\\0&0&1&-1\\0&0&0&0\\\end{matrix}\right]\).
The bottom row becomes all zeros, but there is still a solution for this matrix. I'm assuming this means the vector is in the subspace generated by these vectors.