1
MAT211
Image and Kernel of a Linear Transformation
Domain: (Subset of)R
2
Target: (Subset of )R
4
(0,0)
(1,2)
(8,16)
(0,0,0,0)
(1,2)
(2,2,3,0)
(0,5)
(0,1)
(10,10,15,0)
(0,0,0,4)
F
De±nition
The image of a function f : X > Y is the subset
of elements y of Y which are of the form f(x)
for some x in X.
The image of a function f is denoted by im(f).
Example
•
Describe the image of the linear
transformation f(x,y)=(3x+6y,x+2y)
De±nition
Let v
1
, v
2
,.., v
m
be in R
n
. The span of v
1
, v
2
,.., v
m
is the set of all linear combinations
c
1.
v
1
+ c
2.
v
2+
...+ c
m
v
m
We denote it by span(v
1
, v
2
,.., v
m
)
Question
Consider two vectors v and w in R
n
. Describe
geometrically span(v) and span(v,w).
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Theorem
The image of a linear transformation T(x)=Ax
is the span of the columns of A.
(Compare this theorem with the previous
Example)
Theorem: Properties of
the Image
Consider a linear transformation T from R
m
to
R
n
The zero vector is in the image.
If x and y are in the image, then x+y is in the
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 Fall '08
 MOVSHEV
 Linear Algebra, Vector Space, linear transformation

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