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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 : vector space
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)  0, v V
T :V W
Identity transformation:
T ( v)  v, v  V
T :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 transformati 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.
Then the set of all vectors w in W that are images of vectors
in V is called the range of T and is denoted by 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 the L.T. given by T (x)  Ax, then
rank (T )  rank ( A)
nullity (T )  nullity ( A)
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• One-to-one:
A function T : V  W is called one - to - one if the preimage of
every w in the range consists of a single vector.
T is one - to - one iff for all u and v inV, T (u)  T ( v)
implies that u  v.
one-to-one
not one-to-one
• Onto:
A function T : V  W is said to be onto if every element
in w has a preimage in 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 )  (2 x1  x2  x3 , x1  3x2  2 x3 ,3x2  4 x3 )
 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  R n
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 ' 
( matrix of T relative to B)
(matrix of T relative to B' )
( transitio n matrix from B' to B )
( transitio n matrix from B 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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