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Definition

We got a glimpse of linear transformations when we looked at the correspondence between elementary row operations and elementary matrices. We now look at linear transformations more generally.

Let V and W be vector spaces with scalars coming from the same field F. A mapping T:VW is a linear transformation if for any two vectors x and y in V and any scalar αF, the following are satisfied: T(x+y)=T(x)+T(y)T(αx)=αT(x)

A couple of consequences of this definition are:

  1. T(α1x1++αmxm)=α1T(x1)++αmT(xm) for any vectors x1,,xmV and scalars α1,,αmF.

  2. T(0V)=0W where 0V denotes the zero vector in V and 0W denotes the zero vector in W.

Examples

  1. Let AFm×n. Then T(x)=Ax is a linear transformation from Fn to Fm. To see this, we need to check the two conditions. Let x,yFn and αF. Then T(x+y)=A(x+y)=Ax+Ay=T(x)+T(y) and T(αx)=A(αx)=α(Ax)=αT(x).

  2. Let T be a mapping from P2, the vector space of polynomials in x with real coefficients having degree at most 2, to R2 given by T(ax2+bx+c)=[a+3ca]. For example, T(2x2+1)=[2+3(1)2]=[52] since 2x2+1 can be written as 2x2+0x+1.

    To check that T is a linear transformation, take two vectors a1x2+b1x+c1 and a2x2+b2x+c2 from P2 and a real number γ. Then, T((a1x2+b1x+c1)+(a2x2+b2+c2))=T((a1+a2)x2+(b1+b2)x+(c1+c2))=[(a1+a2)+3(c1+c2)a1+a2]=[(a1+3c1)+(a2+3c2)a1+a2]=[a1+3c1a1]+[a2+3c2a2]=T(a1x2+b1x+c1)+T(a2x2+b2x+c2), and T(γ(a1x2+b1x+c1))=T((γa1)x2+(γb1)x+γc1)=[γa1+3(γc1)γa1]=γ[a1+3(c1)a1]=γT(a1x2+b1x+c1).

Quick Quiz

Exercises

  1. Let T:R5R be such that T(x)=0 for all xR5. Is T a linear transformation? Explain your answer.  

  2. Let T:R3R2 be a linear transformation such that T([102])=[14] and T([111])=[02]. What is T([415])?