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

[vip_snoopy]

Homework Statement

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

Homework Equations

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

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?

 

[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.

 

[vip_snoopy]

strange, my edit was not posted.

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

 

[HallsofIvy]

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.