Topic:3
Linear Transformations
(Linear Maps)
Gyalpozhing College of Information Technology
Prepared by Chiranjeevi Adhikari
Contents
1 Linear Transformation 2
1.1 Kernel and Image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Composition /Product of linear maps . . . . . . . . . . . . . . . . . . . . 5
1.3 Matrix Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Rotation Matrix in R2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
LINEAR ALGEBRA CSM 201
1 Linear Transformation
Two frequently used synonyms for the word function are mapping and transformation.
In Linear Algebra, the domain and range are vector spaces and the word transformation is
preferred.
A transformation T is a rule of correspondence between two vector spaces, a domain V
and a range W, that assigns to each element in V exactly one element( but not necessarily
a different one) in W. Such a transformation is denoted by T : V → W
Definition 1 (Linear Transformation) Let V and W be a vector spaces and a transformation
T : V → W (That is, for each vector v ∈ V, T (v) denotes exactly one vector of W.) is linear
if for any scalars α and any two vectors , u and v in V, the following equality holds:
1. T (u + v) = T (u) + T (v)
2. T (αu) = αT (u)
Example:
The transformation T : R2 → R2 defined by T [a, b] = [a, −b] for all real numbers a and b
is linear.
This transformation maps 2-tuples into 2-tuples by changing the sign of the second component.
Here, T [4, 2] = [4, −2], T [−1, −4] = [−1, 4],and T [−1, 0] = [1, 0].
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LINEAR ALGEBRA CSM 201
In general, setting u = [a, b] and v = [c, d], we have T (u) = [a, −b], T (v) = [c, −d], and
T (u + v) = T ([a, b] + [c, d])
= T ([a + c], [b + d])
= [a + c, −(b + d)]
= [a − b], [c, −d]
= T (u) + T (v)
For any scalar α, we have
T (αu) = T (α[a, b])
=T(αa, αb)
=[αa, −αb]
=α[a, −b]
=αT (u)
Hence , T is linear.
Question:
Determine whether the transformation T is linear if P3 → P2 is defined by
T (a3 t3 + a2 t2 + a1 t + a0 ) = 3a3 t2 + 2a2 t + a1
1.1 Kernel and Image
Definition 2 (Kernel) Let T : U → V be a linear mapping.The kernel of T, written as
kerT,is the set of elements in U that maps into the zero vector 0 in V .i.e
KerT = {u ∈ U, T (u) = 0}
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LINEAR ALGEBRA CSM 201
Definition 3 (Image) The image of T ,written as Im T ,is the set of image points in V.i.e,
ImT = {v ∈ V, ∃u ∈ U for which T (u) = v}
Theorem 1 Let T : U → V be a linear mapping.Then the kernel of T is a subspace of U
and the image of T is a subspace of V.
Example:
Let T : R3 → R3 be the projection of a vector v in xy-plane. i.e,T(x,y,z)=(x,y,0). Find
ker(T) and Im(T).
(Ans:ImT={(a, b, c)|c = 0(xy − plane)} and KerT {(a, b, c)|a = 0, b = 0(z − axis)}
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LINEAR ALGEBRA CSM 201
Question
v2
Let T : R2 → R2 defined by T (v) = −3v1
2v1 + 2v2
1. Find ker(T)
" #
v
2. If A = 1 is the subspace of R2 ,then find the image T (A).
0
1.2 Composition /Product of linear maps
Definition 4 Consider two mapping T : U → V and S : V → W .The composition of T
and S , denoted by S ◦ T ,is a mapping S ◦ T : U → W defined by S ◦ T (a) = S (T (a))
Note: In composition of linear map ,first we apply T to a ∈ U, and then we apply S to
T (a) ∈ V to get S (T (a)) ∈ W
Examples:
Find S ◦ T for T : R3 → R2 , defined by T (x1 , x2 , x3 ) = (x1 + x2 , x2 + x3 ) and S : R2 → R,
defined by S (x1 , x2 ) = (x1 − x2 )
(Ans:S ◦ T (x1 , x2 , x3 ) = x1 − x3 )
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LINEAR ALGEBRA CSM 201
a + b
" # 0
a b
Question Show S ◦ T for T : M2×2 → R4 , defined by S = and S : R4 → P3
c d c + d
0
.a
b
defined T = b + dx2 + (b + d)x3
c
d
Theorem 2 Let U,V,W be vector space .Let T : U → V and S : V → W be linear map or
transformation . Then the composite map S ◦ T is also a linear map.
1.3 Matrix Transformation
Definition 5 Let T : Rn → Rm , for each x in Rn ,T (x) is computed as Ax,where A is an
m × n matrix.
Observation:
1. The domain of T is Rn where A has n columns
2. The co domain of T is Rm when each column of A has m entires
3. The range of T is the set of all linear combinations of the columns of A, because
each image T (x) is of the form Ax
Note: We can write the linear transformation T in matrix transformation Ax
Example ( David clay textbook,pg-74)
1. Find the matrix transformation for T : R2 → R3 defined by T [x1 , x2 ] = [x1 −
3x2 , 3x1 + 5x2 , −x1 + 7x2 ]
1 −3 " x #
(Ans: 3 5 1 )
x2
−1 7
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LINEAR ALGEBRA CSM 201
" # 3
2
2. Let u = and b = 2 ,then
−1
−5
(a) Find T(u),the image of u under the transformation T.
(b) Find x in R2 whose image under T is b.
1.4 Rotation Matrix in R2
" #
cosθ −sinθ
Definition 6 For A = and for any position vector v ∈ R2 , the product Ax is
sinθ cosθ
the vector resulting when x is rotated counter clockwise around the origin by angel θ
Proof:
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LINEAR ALGEBRA CSM 201
cosϕ = Xo
r
,X0 = rcosϕ
sinϕ = Yo
r
, Y0 = rsinϕ
X1 = rcos(ϕ + θ)
’
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LINEAR ALGEBRA CSM 201
Y1 = rsin(ϕ + θ)
Since , X1 = rcos(ϕ + θ)
’
X1 = r(cosϕcosθ − sinϕsinθ)
= rcosϕcosθ − rsinϕsinθ
= X0 cosθ − Yo sinθ (since , ,X0 = rcosϕ and Y0 = rsinϕ )
And Y1 = rsin(ϕ + θ)
Y1 = r(sinϕcosθ + cosϕsinθ)
= rsinϕcosθ + rcosϕsinθ
= Y0 cosθ + X0 sinθ (since , ,X0 = rcosϕ and Y0 = rsinϕ )
Now X1 = X0 cosθ − Y0 sinθ
Y1 = X0 sinθ + Y0 cosθ
" # " #" #
X1 cosθ −sinθ X0
=
Y1 sinθ cosθ Y0
Question
" #
2
Take the point and rotate counter-clockwise by 90o or π2 .
1
