Pls help with linear algebra

Let L: R^3---------------->R^3 be linear transformation defined by L ( [x] )= [-1 2 0] [x]
[y] [1 1 1] [y]....
[z] [2 -1 1] [z]
Is w = [ 1 ] in range of L ???
[ 2 ]
[ -1]

...(|x|).......|-1..2..0|.|x|.......|.1|
L|(y|)..= ..|.1..1..1|.|y|...=..|.2|
...(|z|).......|.2.-1..1|.|z|.......|-…

-x + 2y = 1.............(1)
x + y + z = 2.........(2)
2x - y + z = -1........(3)

If we multiply equation(2) by -1 and add to equation(3) we obtain:

x - 2y = -3

If we now add this equation to equation(1):

x - 2y = -3
-x + 2y = 1
----------------
0 ≠ -2

Therefore, w is not in the range of L.

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