# Show that the range of the linear operator is not all of R^3

1. Jul 14, 2008

### vip_snoopy

1. The problem statement, all variables and given/known data
Show that the range of the linear operator defined by the equations is not all of R3, and find a vector that is not in the range

2. Relevant equations
w1 = x - 2y + z
w2 = 5x - y + 3z
w3 = 4x + y + 2z

3. The attempt at a solution
can I just show that it does not have an inverse and it's a singular matrix to say it's equivalent to the range is not all or R3?

2. Jul 14, 2008

### quasar987

Yes, but you will still have to find a vector that is not in the range. So just find such a vector and solve both problems at once instead.

3. Jul 14, 2008

### vip_snoopy

strange, my edit was not posted.

My question is actually how to find the vector not in the domain?

4. Jul 14, 2008

### HallsofIvy

Staff Emeritus
Yes, that is the whole point.

For what values of w1, w2, w3 does
w1 = x - 2y + z
w2 = 5x - y + 3z
w3 = 4x + y + 2z
NOT have a solution.

Just start solving for x, y, and z and see if you run into a problem (probably dividing by 0) for some values of w1, w2, and w3.