Image and Kernel of a Linear Transformation
Image and Kernel of a Linear Transformation
Image and Kernel of a Linear Transformation
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Example. Find the kernel <strong>of</strong> the linear trans-<br />
formation T from R3 to R2 given by<br />
<br />
Solution<br />
T (x) =<br />
1 1 1<br />
1 2 3<br />
We have to solve the linear system<br />
rref<br />
<br />
⎡<br />
⎢<br />
⎣<br />
T (x) =<br />
<br />
1 1 1 0<br />
1 2 3 0<br />
x1<br />
x2<br />
x3<br />
1 1 1<br />
1 2 3<br />
<br />
=<br />
<br />
<br />
x = 0<br />
x1 − x3 = 0<br />
x2 + 2x3 = 0<br />
⎤<br />
⎥<br />
⎦ =<br />
⎡<br />
⎢<br />
⎣<br />
t<br />
−2t<br />
t<br />
⎤<br />
1 0 −1 0<br />
0 1 2 0<br />
⎡<br />
⎥ ⎢<br />
⎦ = t ⎣<br />
The kernel is the line spanned by<br />
1<br />
−2<br />
1<br />
⎡<br />
⎢<br />
⎣<br />
⎤<br />
⎥<br />
⎦<br />
1<br />
−2<br />
1<br />
<br />
⎤<br />
⎥<br />
⎦.<br />
8
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