Linear Transformation or Linear mapping:
Let V and U be two vector spaces over the same scalar field K .A mapping
F: V U is called as linear mapping or Linear Transformation if
it satisfies the following property:
1) for any vectors v,w ∈ V f( v+w) = f(v) + f(w)
2) for any scalar k and v ∈ V f(kv)=k f(v)
Alternate
For v,w in V and for any scalar c1 , c2
F( c1 v+c2w)= c1 F(v)+c2F(w) then F is linear mapping.
Let A : R^n − − −→ R^n be a linear Transformation
Properties of Linear Transformation:
1. Every square matrix defines a Linear Transformation
2. Singular transformation: If |A|=0 then transformation Y = AX is called as singular transformation
3. Non Singular transformation: If |A| not equal to 0 then transformation Y = AX is
called as non singular transformation or Regular transformation.
4. Orthogonal transformation: If A is orthogonal matrix then Y =AX is called as Orthogonal transformation.
Matrix A is orthogonal if and only if AA| = A|A = I (Identity Matrix)
5. A non singular Linear Transformation carries linearly independent vectors into linearly independent vectors.
6. Composite Transform: If A: X−−→ Y & B : Y −−→ Z any two transforms then Linear Transformation AB: X : −−→ Z is called as composite Transformation
[If X=AY and Y = BZ then composite Transformation X= (AB) Z
expresses X in terms of Z]
Example:
Let f : R^3---------->R^2 defined as f(( x,y,z)) = (x+y, y-z)
Does f is a linear Transformation?
--->1) for v,w ∈ R^3 f(v+w) = f(v)+f(w)
For v=(a,b,c) and w= (p,q, r) in R^3
L H S = f(v+w) = f( ( a b c) + ( p q r) ) = f( ( a+p, b+q, c+r) = ( a+p+b+q, b+q-c-r)
RHS= f(v)+f(w) = f( (a,b,c) ) +f( (p,q,r) = (a+b, b-c)+( p+q, q-r) = ( a+b+p+q, b-c+q-r)
RHS = ( a+b+p+q, b+q-c-r)
Thus LHS = RHS
f(v+w) = f(v)+f(w)
2) for any scalar k
f(kv) = k f(v)
F(kv)= f( k( a,b,c) )= f( ( ka,kb,kc) ) = ( ka+kb, kb-kc)= k ( a+b, b-c)
= k f( a b c) = k f( V)
Thus f(kv)= k f(v)
From 1) and 2) f is linear Transformation.
Note : Under Linear Transformation zero vector of Space v is mapped to zero
vector of space U:
If T is linear Transformation then T(kv) = k T(v) for all scalar k
For k=0
T( 0v)= 0 T(v)
T(0)= 0
Example 2) If F( x,y,z) = (x+1, y+3, z-2)
--->F(0 ,0, 0)= ( 0+1,,0+3, 0-2)= ( 1,3, -7)
Zero is not mapped to zero therefore F is not a linear Transformation.
Example 3)
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