How to Determine a Vector Not in the Range of a Linear Transformation?

[georgeh]

The question asls:Show that the range of the linear operator defined by the equations:
W_1=x_1 - 2*x_2 + x_3
W_2=5*x_1-x_2+3 *x_3
W_3=4*x_1+x_2+2*x_3
is not all of R^3, and find a vector that is not in the range.
Well, we know T
T=[1,-2,1;5,-1,3;4,1,2]
I augment W
and we get
T|W=[1,-2,1,W_1;5,-1,3,W_2;4,1,2,W_3]
I do Reduce Echelon
I get zeros on the bottom
so i get
w_1-W_2+W_3=0
k, well, i chose a vector point, (10,5,6)
i get
10 != -1
not sure if this is correct way of solving it.

 

[daveb]

If I read your problem correctly, here's a hint. If there is a vector that is not in the range of the transformation, then what does that tell you about the linear dependence or independence of the three equations?

 

[georgeh]

we haven't covered linear independence yet.

 

[HallsofIvy]

The question asls:Show that the range of the linear operator defined by the equations:
W_1=x_1 - 2*x_2 + x_3
W_2=5*x_1-x_2+3 *x_3
W_3=4*x_1+x_2+2*x_3
is not all of R^3, and find a vector that is not in the range.
Well, we know T
T=[1,-2,1;5,-1,3;4,1,2]
I augment W
and we get
T|W=[1,-2,1,W_1;5,-1,3,W_2;4,1,2,W_3]
I do Reduce Echelon
I get zeros on the bottom
so i get
w_1-W_2+W_3=0
k, well, i chose a vector point, (10,5,6)
i get
10 != -1
not sure if this is correct way of solving it.

WHY did you choose (10, 5, 6) and HOW did you "get 10 != -1"?