Chapter 6
Linear Transformations
6.1 Introductions to Linear Transformations
• Function T that maps a vector space V into a vector space W:
T : V mapping
W ,
V ,W : vectorspace
V: the domain of T
W: the codomain of T
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• Image of v under T:
If v is in V and w is in W such that
T ( v)  w
Then w is called the image of v under T .
• the range of T:
The set of all images of vectors in V.
• the preimage of w:
The set of all v in V such that T(v)=w.
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• Notes:
(1) A linear transformation is said to be operation preserving, because
the same result occurs whether the operations of addition and
scalar multiplication are performed before or after T.
T (u  v)  T (u)  T ( v)
Addition
in V
Addition
in W
T (cu)  cT (u)
Scalar
multiplication
in V
Scalar
multiplication
in W
(2) A linear transformation T : V  V from a vector space into
itself is called a linear operator.
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• Two simple linear transformations:
Zero transformation:
T :V  W
T ( v)  0, v V
Identity transformation:
T :V  V
T ( v)  v, v V
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6.2 The Kernel and Range a Linear
Transformation
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• Note:
The kernel of T is sometimes called the nullspace of T.
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T (x)  Ax (a linear tr ansformati on T : R n  R m )
 Ker (T )  NS ( A)  x | Ax  0, x  R m  (subspace of R m )
• Range of a linear transformation T:
Let T : V  W be a L.T .
T hen theset of all vectorsw in W thatare images of vectors
in V is called therange of T and is denotedby range(T )
range(T )  {T ( v) | v V }
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• Notes:
T : V  W is a L.T.
(1) Ker(T ) is subspace of V
(2)range(T ) is subspace of W
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• Note:
Let T : R n  R m be theL.T .given by T (x)  Ax, then
rank(T )  rank( A)
nullity(T )  nullity( A)
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• One-to-one:
A functionT : V  W is called one- to - oneif thepreimageof
everyw in therange consistsof a single vector.
T is one- to - oneiff for all u and v inV, T (u)  T ( v)
implies thatu  v.
one-to-one
not one-to-one
• Onto:
A functionT : V  W is said to be ontoif everyelement
in w has a preimagein V
(T is onto W when W is equal to the range of T.)
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6.3 Matrices for Liner Transformations
• Two representations of the linear transformation T:R3→R3 :
(1)T ( x1, x2 , x3 )  (2x1  x2  x3 , x1  3x2  2x3 ,3x2  4x3 )
 2 1  1  x1 
(2)T (x)  Ax   1 3  2  x2 

 
0
3
4

  x3 
• Three reasons for matrix representation of a linear
transformation:
– It is simpler to write.
– It is simpler to read.
– It is more easily adapted for computer use.
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• Notes:
(1) The standard matrix for the zero transformation from Rn into Rm
is the mn zero matrix.
(2) The standard matrix for the identity transformation from Rn into
Rn is the nn identity matrix In
• Composition of T1:Rn→Rm with
T2:Rm→Rp :
T ( v)  T2 (T1 ( v)), v  Rn
T  T2  T1
domain of T  domain of T1
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• Note:
T1  T2  T2  T1
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• Note: If the transformation T is invertible, then the inverse
is unique and denoted by T–1 .
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6.4 Transition Matrices and Similarty
T :V  V
( a L.T ).
B  {v1 , v2 ,, vn } ( a basis of V ), B'  {w1 , w2 ,, wn } (a basis of V )
A  T (v1 )B , T (v2 )B ,, T (vn )B 
A'  T (w1 )B' , T (w2 )B' ,, T (wn )B' 
P  w1 B , w2 B ,, wn B 
P1  v1 B' , v2 B' ,, vn B' 
( matrixof T relativeto B)
(matrixof T relativeto B' )
( transition matrixfromB' to B )
( transition matrixfromB to B' )
 v B  Pv B ' ,
vB '  P 1vB
T ( v)B  AvB
T ( v)B'  A' vB'
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• Two ways to get from
vB' to T (v) :
B'
indirect
(1)(direct)
A'[ v]B '  [T ( v)]B '
(2)(indirect)
P 1 AP[ v]B '  [T ( v)]B '
 A'  P 1 AP
direct
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• Note: From the definition of similarity it follows that any tow
matrices that represent the same linear transformation
T : V  V with respect to different based must be similar.
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6.5 Applications of Linear Transformations
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