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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Definition. Consider the vectors v1, v2, . . . ,<br />
vn in R m . The set <strong>of</strong> all linear combinations <strong>of</strong><br />
the vectors v1, v2, . . . , vn is called their span:<br />
span(v1, v2, . . . , vn)<br />
={c1v1 + c2v2 + . . . + cnvn: ci arbitrary scalars}<br />
Fact The image <strong>of</strong> a linear transformation<br />
T (x) = Ax<br />
is the span <strong>of</strong> the columns <strong>of</strong> A. We denote<br />
the image <strong>of</strong> T by im(T ) or im(A).<br />
Justification<br />
T (x) = Ax =<br />
⎡<br />
⎢<br />
⎣<br />
| |<br />
v1 . . . vn<br />
| |<br />
= x1 v1 + x2 v2 + . . . + xn vn.<br />
⎡<br />
⎤<br />
⎢<br />
⎥ ⎢<br />
⎦ ⎢<br />
⎣<br />
x1<br />
x2<br />
.<br />
xn<br />
⎤<br />
⎥<br />
⎦<br />
3