2. Linear transformation is a function that converts one data type into another type of data.
Linear transformation from 𝑹 𝒏 to 𝑹 𝒎 is referred to as Euclidean linear transformation
whereas, linear transformation from vector space V to vector space W is referred to as
general linear transformation.
Let V and W be a vector spaces. A L.T. (T:v->w) is a function T from V to W such that
(a)T(u+v)=T(u)+T(v)
(b)T(ku)=kT(u)
If above conditions satisfy thant it is linear transformation.
For all vectors u and v in V and all scalars k.
If V=W, the linear transformation T:V->V is called a linear operator.
3. Examples:
(1)T:𝑹 𝟐 → 𝑹 𝟐 T(x,y)=(x+y,x-y)
Let u=(𝑥1, 𝑦1) and v=(𝑥2, 𝑦2) be a vectors in 𝑹 𝟐 and k be any scaler.
u+v=(𝑥1+𝑥2, 𝑦1+𝑦2)
T(u+v) = (𝑥1+𝑥2+𝑦1+𝑦2, 𝑥1+𝑥2-𝑦1-𝑦2)
=(𝑥1+𝑦1, 𝑥1-𝑦1)+(𝑥2+𝑦2, 𝑥2-𝑦2)
=T(u)+T(v)
Ku=(𝑘𝑥1, k𝑦1)
T(ku)= (𝑘𝑥1+k𝑦1, k𝑥1-k𝑦1)
= k(𝑥1+𝑦1, 𝑥1-𝑦1)
=kT(u)
Hence , It is linear transformation
4. (2)T:𝑀 𝑚𝑛->R ,T(A)=det(A)
Let 𝐴1and 𝐴2 be a two matrices with same order in 𝑀 𝑚𝑛.
T(𝐴1+𝐴2)=det(𝐴1+𝐴2)
≠det(𝐴1)+det(𝐴2)
≠T(𝐴1)+T(𝐴2)
Hence,It is not linear transformation.
A linear transformation is completely determined by the images of any set of basis vectors.If
T:V->W is alinear transformation if {𝑣1, 𝑣2,…, 𝑣 𝑛} is any basis for Vthen any vector v in V is
expressed as a linear combination of 𝑣1, 𝑣2,…, 𝑣 𝑛.
v=𝑘1 𝑣1+𝑘2 𝑣2,…, 𝑘 𝑛 𝑣 𝑛
The L.T. T(v) is given by,
T(v)=T(𝑘1 𝑣1+𝑘2 𝑣2,…, 𝑘 𝑛 𝑣 𝑛) = 𝑘1 𝑇(𝑣1)+𝑘2 𝑇(𝑣2),…, 𝑘 𝑛 𝑇(𝑣 𝑛)
5. (3)Consider the basis S={𝑣1, 𝑣2} 𝑎𝑛𝑑 𝑣1=(1,1) , 𝑣2=(1,0) and let T:𝑹 𝟐 → 𝑹 𝟐 be the L.T. such
that T(𝑣1)=(1,-2) and T(𝑣2)=(-4,1).Find formula for T(𝑥1, 𝑥2) and also find T(5,-3).
Let v=(𝑥1, 𝑥2) be an arbitrary vector in 𝑹 𝟐 and can be expressed as a linear combination of
𝑣1 𝑎𝑛𝑑 𝑣2.
(𝑥1, 𝑥2) =𝑘1(1,1)+𝑘2(1,0)
=(𝑘1+𝑘2 , 𝑘1)
Comparing both sides,
𝑘1+𝑘2=𝑥1
𝑘1=𝑥2
Therefore,
𝑘1=𝑥2
𝑘1=𝑥1- 𝑥2
∴ v=𝑥2 𝑣1+(𝑥1-𝑥2) 𝑣2
Now,
T(𝑥1, 𝑥2)=𝑥2 𝑇(𝑣1)+(𝑥1-𝑥2)T(𝑣2)
6. =𝑥2(1,-2)+(𝑥1-𝑥2)(-4,1)
=(-4𝑥1+5𝑥2 , 𝑥1-3𝑥2)
T(5,-3)=(-4(5)+5(-3),5-3(-3))
=(-35,14)
Let V and W be two vector spaces and let T:V->W, be a linear transformation. The kernel or null
space of T, denoted as ker(T) or N(T),is the set of all vectors in V that T maps into the zero vector
The range of T, denored by R(T), is the set of all vectors in W that are images of at least one
vector in V under T.
Theorem:
If T:V->W is a linear transformation then
1)The kernel T is a subspace of V.
2)The range of T is a subspace of W.
7. If T:V->W is a linear transformation from afinite dimensional vector space V to a vector space W
then
rank(T)+nullity(T)=dim V
(4) Let T:𝑃2 → 𝑅2 be a linear transformation defined by
T(𝑎0+𝑎1 𝑥+𝑎2 𝑥2)=(𝑎0-𝑎1, 𝑎1+𝑎2)
1)Find a basis of ker(T).
2)Find a basis of R(T).
3)Verify the dimension theorem.
1)—
The basis for ker(T) is the basis for the solution space of the homogeneous system:
𝑎0-𝑎1=0
𝑎1+𝑎2=0
Let,
8. 𝑎2=t
𝑎1=-t
𝑎0=-t
𝑎0
𝑎1
𝑎2
=
−𝑡
−𝑡
𝑡
=t
−1
−1
1
=t𝑣1
Hence basis for ker(T)= {𝑣1}= {-1-𝑥 +𝑥2}
2)-
The basis for the range of T is the basis for the columm space of [T]
[T]=
1 −1 0
0 1 1
The leading 1’s appear in columns 1 and 2.
Hence,
basis for R(T)=basis for column space of [T]
10. Here diagram is self explanatory.
In other words we can say that for any Linear transformation T:V->W ,
1)It is one-one if it has only one kernel element i.e,Zero element.
2)It is onto if Range(T)=W.
Let T:V->W be a linear transformation, it is isomorphism it is isomorphism
between V and W if it is both one-one and onto.
Here T is called Bijective Transformation.
11. (5)T:𝑅2 → 𝑅2, where T(x,y)=(x+y , x-y)
Check T for one-one,or both or neither.
T(x,y)=(0,0)
(x+y,x-Y)=(0,0)
x+y=0
X-y=0
Solving above equation we get,
(x,y)=(0,0).
So ker(T)={0}
Hence, T is one-one.
12. A linear transformation is onto if R(T)=W
Let v=(x,y) and w=(a,b) be in 𝑅2, where a and b are real numbers such that T(v)=W.
T(x,y)=(a,b)
(x+y,x-y)=(a,b)
x+y=a
x-y=b
Solving above equations we get,
x=(a+b)/2
y=(a-b)/2
Thus for evert w=(a,b) in 𝑅2
, there exists a V=((a+b)/2,(a-b)/2) in 𝑅2
.
Hence T is onto.
13. Here T is isomorphism between 𝑅2
and 𝑅2
because it is both one-one and onto linear
transformation.
Let T:V->W be a linear transformation of an n-dimensional vector space V. to an m-dimensional
vector space W.(where n≠ 0 and m≠0) and let 𝑆1 = {𝑣1, 𝑣2, … , 𝑣 𝑛} and 𝑆2 = {𝑤1, 𝑤2,…, 𝑤 𝑛} be
bases for V and W respectively.
If A is the standard matrix of this transformation then
A=[[T(𝑣1) 𝑆2
] [T(𝑣2) 𝑆2
] … [T(𝑣 𝑛) 𝑆2
]]
(6)
14. Standard basis for 𝑀33 are S={
1
0
0
,
0
1
0
,
0
0
1
}
Their linear transformations are
1
0
,
−1
0
,
0
2
respectively.
So the Standard matrix will be:
A=
1 − 1 0
0 0 2
15. A specific application of linear transformation is for geometric transformations, such as those
performed in computer graphics, where the translation, rotation and scaling of 2D or 3D
objects is performed by the use of a transformation matrix. Linear transformations also are
used as a mechanism for describing change: for example in calculus correspond to derivatives;
or in relativity, used as a device to keep track of the local transformations of reference frames.
Another application of these transformations is in compiler optimizations of nested-loop code,
and in parallelizing compiler techniques.
