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Linear Transformations part 3-Algebra-Lecture Slides, Slides of Algebra
This lecture was delivered by Dr. Kunjabihari Negi at Aligarh Muslim University for Algebra course. It includes: Linear, Transformation, Real-valued, Functions, matrix, Multiplication, Vector, Generalization, Domain, Range
Typology: Slides
2011/2012
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Example
Example
Example (Contd.)
The Matrix of a LinearTransformation ^ Every Linear Transformation
T:R
n^ -> R
m^ is a matri
transformation
x -> Ax
^ The key to finding
A^
is to observe that
T^ is completel
determined by what it does to the columns of the n×identity matrix
I^. n
SolutionWrite Since T is a linear transformation
1
1
2
1 1
2 2
2
x
x^
x^
x^
x e^
x e
^ x
^
^
^
^
^
^
^
2
2
1
2
1
2
1
2 1
( )^
(^ )^
(^ ) 5
3
5
3
7
8
7
8
2
0
2
0
T x x T e
x T e
x^
x
x^
x^
x^
x x
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
The Matrix of a LinearTransformation – Example 1
^
^
1
1
2
2
( )
(^ )
(^
)^
x
T x
T e
T e
Ax ^ x
^
^
^
The Matrix of a LinearTransformation – Example 1^ Thus
5
3 7
8 2
^0 ^
^
^ ^
^
^
A
ProofWrite x=I
x=[en
… e 1
]x = xn
e+…+x 11
enn , and using the Linearity
of T to compute The^ matrix
A^
is^ called
the
Standard
Matrix
for
the
Linear
Transformation
T.
^
1 1^
1
1 1
1
( )^
(^
)^
(^ )
(^ )
(^ )
n^ n^ (^ )
n^
n
n
n
T x
T x e
x e^
x T e
x T e
x
T e
T e
Ax
x
^
^
^
^
^
^
^
^
^
^
^
^
^
The Matrix of a LinearTransformation
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^ Find
the
standard
matrix
A
for
the
dilation
transformation T(x) = 3x, for x in R
SolutionWrite
1
1
2
2
3
0
(^ )^
3
and
(
)
3
0
3
T e
e^
T e
e
^
^
^
^
3
0 0
3
A^
^
^
The Matrix of a LinearTransformation – Example 1
The Matrix of a LinearTransformation – Example 2
A rotation transformation
Onto mapping ^ A mapping T:R
n^ -> R
m^ is said to be onto R
m^ if each
in R
m^ is the image of
at least
one
x^ in R
n.
One-to-one mapping^
A mapping T:R
n^ -> R
m^ is said to be one-to-one if eac
b^ in R
m^ is the image of
at most
one
x^ in R
n.
One-to-one mapping - Example ^ Let T be the linear transformation whose standard matriis SolutionA happens to be in echelon form, we can see at once tha
A has a pivot position in each row.
For each
, the equation Ax=b is consistent. In other
words, the linear transformation T maps ( its domain)onto
However, since the equation Ax=b has free
variables, each b is the image of more than one x , i.e., T
is^ not one-to-one
.
(^4) R
1
4
8
1
0
2
1 3
0
0
0
5
A
^
^
^
^
^
^
3 R
b^ ^3^ R
Onto and one-to-one mapping Theorem
Let T:R
n^ -> R
m^ be a linear transformation and let A
be the standard matrix for T, then:^ a)
T maps R
n^ onto R
m^ if and only if the columns of A
span R
m
b)^ T is one-to-one if and only if the columns of A arelinearly independent Proofa. The columns of A span R
m^ if and only if for each
b^ the
equation
Ax=b
is consistent.
Onto and one-to-one mapping ^ In other words, if and only if for every
b , the equation
T(x)=b
has at least one solution. This is true only if and only if
T^ maps
n R onto
m R
b. The equations
T(x)=
and
Ax=
are similar except
notation.
T^
is one-to-one if only if
Ax=
has only the
trivial solution. This happens only if the columns of
A
are linearly independent.
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