Linear Algebra: UNIT II: Linear Transformations and Diagonalization
Definition and Properties of Linear Transformation
Definition and Properties of Linear Transformation
LINEAR TRANSFORMATION
Definition:
Let
V and W be vector spaces over F. A function
T: V→W a linear transformation from V to W if
for all x, y ∈
V and c ∈ F, we have
(a)
T(x+y) = T(x) + T (y)
(b)
T(cx)=cT(x))
Properties
Property 1: If T is
linear then T (0)=0
Proof:
We
know that T(x+y)=T(x)+T (y) and
T(0)
= T (0+0) = T (0) + T (0)
=
0 and
T(0)=T(0x)=0T
(x)=0.
Property 2: T is linear
if and only if
T(cx+y) = cT(x) + T(y); for all x, y ∈ V and c ∈ F.
Proof:
Assume
T is linear
T
(cx+y) = T (cx) + T (y)
= cT (x)+T (y)
Conversely,
Assume
T (cx + y) = cT (x) + T (y)
put
c=1
T
(x + y) = T (x) + T (y).
y ∈
V; y = 0 ∈
V
y(cx+0)
= cT (x)+T (0)
=
cT(x)+0 = cT(x)
T is linear
Property 3: If T is
linear then T(x−y) = T (x) − T (y) for all x,y ∈ V.
Proof
To
prove T (ax+by) = aT(x)+bT (y) for x, y ∈
V and a, b ∈
F.
The
scalars are considered as 1,‒1 ∈
F, we assume
a = 1 and b = −1
T(x
− y) = (1)T (x) + (− 1) T(y) = T(x) − T(y)
T(x−y)=T(x)−T(v)
Property 4: T is linear
if and only if, x1, x2,
x3... xn ∈
V and a1, a2, aз... an ∈ F then we have
Proof
Given
that T is linear
To
prove T(α1 x1 +
a2 x2 + a3
x3 + ... + anxn)
=
a1 T(x1) + a2 T(x2) + a3T (x3)
+ ... + anT(xn).
From
the concept of linear combination, linear combination of vectors in V is also a
vector in V.
Let
α1 x1 + a2 x2 + a3
x3 + ... + anxn = y ∈ V.
T(а1x1 + а2x2
+ a3x3 + ... + anxn) = T(a1x1 + y)
=
a1T(x1) + T(y)
=
a1T(x1) + T(a2x2
+ a3x3 + ... + ... + anxn)
Similarly
let us consider a3x3
+ α4x4 + …. + anxn = z
T
(α1 x1 + a2x2 + ... + anxn)
= a1T(x1) + T (a2x2
+ z)
=
a1T(x1) + a2T
(x2) + T(z)
Proceeding
in the same way, we get
T(a1
x1 + a2 x2
+ ... + anxn) = a1
T(x1) + a2 T(x2) + ... + an T(xn)
Conversely,
let T (a1 x1 + a2 x2
+ ... + anxn)
=
a1 T(x1) + a2T
(x2) + …. + an T(xn)
It
is applicable for any scalars a1, a2,... an and vectors x1, x2, x3
... xn.
Since
0 ∈ V, a3 = a4 = a5
= ... = a1 = 0 and a1 = a2 = 1.
Then
T (1.x1 + 1.x2
+ 0.x3 + … + 0xn)
=
1 • T (x1) + 1 • T (x2)
+ 0 • T (x3) + ... + 0T (xn)
⇒ T(x1+x2)=T(x1)
+ T(x2)
Similarly,
we have a2 = a3
= a4 = ... = an = 0
then
we get T (a1 x1) = α1T(x1)
T is linear.
Linear Algebra: UNIT II: Linear Transformations and Diagonalization : Tag: maths, mathematics : - Definition and Properties of Linear Transformation
Linear Algebra: UNIT II: Linear Transformations and Diagonalization
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Linear Algebra
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