What Does the Kernel of a Matrix Tell Us About Its Null Space?

[beanryu]

please help
find all x in R4 that are mapped into teh zero vector by A
A =
1 3 9 2
1 0 -4 3
0 1 2 3
-2 3 0 5

i think the problem wants me to find x where x times A = 0

so i did this augmented matrix
A=
1 3 9 2 0
1 0 -4 3 0
0 1 2 3 0
-2 3 0 5 0

then, after row reduction and stuff, i got this

1 0 3 0 0
0 1 2 0 0
0 0 0 1 0
0 0 0 1 0

all i know is x4 = 0... it doesn't seem to give anything about x1 x2 and x3... are their values arbitrary or am I wrong?

thank you.

 

[lippyka]

The values of x1, x2 and x3 are arbitrary. You can assign any values to them and it will still satisfy the equation xA = 0.