Linear Dependence: Is (1,0,0), (-3,7,0) & (1,1,0) Independent?

[forty]

(1,0,0) (-3,7,0) and (1,1,0)

I'm trying to work out if these vectors are linearly independent or not.
Intuitively i believe they are dependent as they span the xy-plane.. but then how do i work out the linear combinations.

e.g:
(1,0,0) = a(-3,7,0) + b(1,1,0) where a and b are real numbers.

Ive tried writing the vectors as column vectors, then row reducing and i end up with

[1 -10 0;0 7 1;0 0 0] (which has rank(2) which means they are linearly dependent)

but as for where to go from here I'm lost.

Any help greatly appreciated.

 

[HallsofIvy]

(1,0,0) (-3,7,0) and (1,1,0)

I'm trying to work out if these vectors are linearly independent or not.
Intuitively i believe they are dependent as they span the xy-plane.. but then how do i work out the linear combinations.

e.g:
(1,0,0) = a(-3,7,0) + b(1,1,0) where a and b are real numbers.

Okay, so 1= -3a+ b and 0= 7a+b. Solve those two equations for a and b. I would recommend subtracting one equation from the other!

Ive tried writing the vectors as column vectors, then row reducing and i end up with

[1 -10 0;0 7 1;0 0 0] (which has rank(2) which means they are linearly dependent)

but as for where to go from here I'm lost.

Any help greatly appreciated.

 

[forty]

Is there a way of doing this with matrices?

So writing the vectors as columns and row reducing, and from here deciding what are the linear combinations?

so [1 -3 1;0 7 1;0 0 0]

becomes [1 -10 0;0 7 1;0 0 0]from here is there a way of getting the answer directly from this matrix?

 

[HallsofIvy]

You prefer a harder way? Writing the two equations, 1= -3a+ b and 0= 7a+ b as an "augmented" matrix gives
$$\left[\begin{array}{ccc}-3 & 1 & 1\\ 7 & 1 & 0\end{array}\right]$$
where, as you can see, the vectors form the rows. That row reduces to
$$\left[\begin{array}{ccc} 1 & 0 & -\frac{1}{10} \\ 0 & 1 & \frac{7}{10}\end{array}\right$$
showing that the equations have a solution (a= -1/10 and b= 7/10) and so the vectors are dependent. As for your initial method, the very fact that your matrix has rank only two tells you that the vectors are dependent.

Writing the

 

[forty]

Once again thank you! you are my new Walter Lewin!