Introduction To Linear Transformation


Linear Algebra / Tuesday, November 6th, 2018
(Last Updated On: November 8, 2018)

Linear Transformation Definition

Let V and W be vector spaces over the field F, a transformation T from V to W is said to be a linear transformation if

\[T\left( \alpha +\beta  \right)=T\left( \alpha  \right)+T\left( \beta  \right)\]

\[and~~~T\left( k\alpha  \right)=kT\left( \alpha  \right)\]

Where α and β are arbitrary elements of V and k is an element of F.

The above two conditions can be replaced by a single condition

\[T\left( k\alpha +\beta  \right)=kT\left( \alpha  \right)+T\left( \beta  \right)\]

 Example 01

If F be a field and let V be the space of polynomial functions f from V into V, given by

\[f\left( x \right)={{c}_{0}}+{{c}_{1}}x+{{c}_{2}}{{x}^{2}}+…+{{c}_{k}}{{x}^{k}}\]

\[Let~~D\left( f\left( x \right) \right)={{c}_{1}}+2{{c}_{2}}x+3{{c}_{3}}{{x}^{2}}+…+k{{c}_{k}}{{x}^{k-1}}\]

The D is a linear transformation from V into V.

 Example 02

For the vector space R2, let the transformation T: R2 → R2 be defined in such a way that

\[T\left( \left( a,b \right) \right)=\left( a+2,b \right)\]

This transformation is not linear, because

\[T\left( \left( {{a}_{1}},{{b}_{1}} \right)+\left( {{a}_{2}},{{b}_{2}} \right) \right)=T\left( \left( {{a}_{1}}+{{a}_{2}},{{b}_{1}}+{{b}_{2}} \right) \right)\]

\[=\left( {{a}_{1}}+{{a}_{2}}+2,{{b}_{1}}+{{b}_{2}} \right)\]

\[whereas~~T\left( \left( {{a}_{1}},{{b}_{1}} \right) \right)+T\left( \left( {{a}_{2}},{{b}_{2}} \right) \right)\]

\[=\left( {{a}_{1}}+2,{{b}_{1}} \right)+\left( {{a}_{2}}+2,{{b}_{2}} \right)\]

\[=\left( {{a}_{1}}+{{a}_{2}}+4,{{b}_{1}}+{{b}_{2}} \right)\]

\[\therefore T\left( \left( {{a}_{1}},{{b}_{1}} \right)+\left( {{a}_{2}},{{b}_{2}} \right) \right)\ne T\left( \left( {{a}_{1}},{{b}_{1}} \right) \right)+T\left( \left( {{a}_{2}},{{b}_{2}} \right) \right)\]

\[Also~~T\left( k\left( a,b \right) \right)=T\left( \left( ka,kb \right) \right)=\left( ka+2,kb \right)\]

\[and~~kT\left( \left( a,b \right) \right)=k\left( a+2,b \right)=\left( ka+2k,kb \right)\]

\[So,~~T\left( k\left( a,b \right) \right)\ne kT\left( \left( a,b \right) \right)\]

Hence the transformation T is not linear.

 Example 03

Let T: R3 → R3 be defined by T((a1, a2, a3))= (a1, a2, 0), (a1, a2, a3) ∈ R3.

Let α = (a1, a2, a3) and β = (b1, b2, b3) ∈ R3.

Then α + β = (a1, a2, a3) + (b1, b2, b3) = (a1 + b1, a2 + b2, a3 + b3)

T(α + β) = (a1 + b1, a2 + b2, 0) = (a1, b1, 0) + (a2, b2, 0) = T(α) + T(β)

And for any c ∈ R,

cα = c(a1, a2, a3) = (ca1, ca2, ca3)

Therefore, T(cα) = T((ca1, ca2, ca3)) = (ca1, ca2, 0) = c(a1, a2, 0) = cT(α)

Hence T is linear transformation.

Image of an element

If T: V → W be a transformation then for any element α in V we get an element α’ in W. We write T(α) = α’. Here α’ is called image of α by T. In other word T(α) is an image of α. If α ∈ V then its image T(α)∈ W.

 Example 04

If T: V2 → V3 is defined as T(x1, x2) = (x1 + x2, x1, x2). Then T(2, 3) = (2 + 3, 2, 3) = (5, 2, 3). So (5, 2, 3) is image of (2, 3).

Image Set

Let T: V → W be a transformation. Set of all images by T is a subset of W. This subset is called image set of T. It is denoted by Im(T) of T(V). Obviously Im(T) ⊂ W.

 Theorem

For a linear transformation T: V → W, Im(T) is a subspace of V.

 Theorem

Let V and W be vector spaces over a field F and T: V → W be a transformation.

Then (i) T(0) = 0’, where 0 and 0’ are null elements in V and W respectively.

(ii) T(-α) = -T(α) for all α ∈ V.

Proof:

In V we have 0 + 0 = 0

Therefore T (0 + 0) = T(0)

But T is linear, so T (0 + 0) = T(0) + T(0)

⇒ T(0) + T(0) = T(0)

⇒ T(0) + T(0) = T(0) + 0’

⇒ T(0) = 0’

Now, α + (-α) = 0 in V

Therefore, T(α + (-α)) = T(0) = 0’

⇒ T(α) + T(-α) = 0’

⇒ T(-α) = -T(α)

Rank of a Linear Transformation

Let T: V → W be a transformation. We have seen the image set T(V) is a subspace of W. The dimension of this subspace T(V) or Im(T) is called the Rank of T.

Identity Transformation

Let V be a vector space. The transformation I: V → V defined by I(α) = α is called identity transformation.

Zero Transformation

Let V and W be two vector spaces. Let θ and θ’ be the two null vectors in V and W respectively. Then the transformation O: V → W defined by O(α) = θ’ for all α in V is called zero transformation

 Theorem

Identity transformation and zero transformation are linear transformation.

 

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