Solving a linear system of equations

[jesuslovesu]

Span of vectors in R3

Homework Statement

u1 = (1,0,-1)
u2 = (1,1,1)
u3 = (3,1,-1)
Determine whether the vectors span R^3.

Homework Equations

The Attempt at a Solution

I know how to determine if the vectors span R^3(or maybe i dont). In this case checking if I can find a linear combo for (1,0,0)

Step 1:
1 = a + b + 3c
0 = 0 + b + c
0 = -a + b - c

Step 2:
b = -c

Step 3:
1 = a - c + 3c
1 = a + 2c

Step 4:
a = b -c
a = -2c

Step 5:
1 = (-2c) + 2c
1 = 0

Now with 1 = 0, I would think that the linear system of equations cannot be solved.

**Solving this problem with my graphing calculator I get the same answer, however my book states that these three vectors do indeed span R^3. Why is that?

 

[Mindscrape]

You are looking for whether or not linear combinations of the vectors equal each other. So

c_1 (1,0,-1)^T + c_2 (1,1,1) + c_3 (3,1,-1) = 0

Can you find any solution to that other than c_1=c_2=c_3=0? If not the vectors are linearly independent, and you have one condition of span down. What is the other?