# 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 R ^{2}, let the transformation T: R^{2} → R^{2} 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: R^{3} → R^{3} be defined by T((a_{1}, a_{2}, a_{3}))= (a_{1}, a_{2}, 0), (a_{1}, a_{2}, a_{3}) ∈ R^{3}.

Let α = (a_{1}, a_{2}, a_{3}) and β = (b_{1}, b_{2}, b_{3}) ∈ R^{3}.

Then α + β = (a_{1}, a_{2}, a_{3}) + (b_{1}, b_{2}, b_{3}) = (a_{1} + b_{1}, a_{2} + b_{2}, a_{3} + b_{3})

T(α + β) = (a_{1} + b_{1}, a_{2} + b_{2}, 0) = (a_{1}, b_{1}, 0) + (a_{2}, b_{2}, 0) = T(α) + T(β)

And for any c ∈ R,

cα = c(a_{1}, a_{2}, a_{3}) = (ca_{1}, ca_{2}, ca_{3})

Therefore, T(cα) = T((ca_{1}, ca_{2}, ca_{3})) = (ca_{1}, ca_{2}, 0) = c(a_{1}, a_{2}, 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: V_{2} → V_{3} is defined as T(x_{1}, x_{2}) = (x_{1} + x_{2}, x_{1}, x_{2}). 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 I_{m}(T) of T(V). Obviously I_{m}(T) ⊂ W.

Theorem |

**For a linear transformation T: V → W, I _{m}(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 I_{m}(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.**