Properties of linear transformation - Linear Transformation

Properties of linear transformation

Lessons

Notes:
Recall from last chapter the 2 properties of AxAx:
1. A(u+v)=Au+AvA(u+v)=Au+Av
2. A(cu)=c(Au)A(cu)=c(Au)

where uu and vv are vectors in Rn\Bbb{R}^n and cc is a scalar.

Now the properties of linear transformation are very similar. Linear transformation preserves the operations of vector addition/subtraction and scalar multiplication. In other words, If T is linear, then:
1. T(u+v)=T(u)+T(v)T(u+v)=T(u)+T(v)
2. T(cu)=cT(u)T(cu)=cT(u)
3. T(0)=0T(\vec{0})=\vec{0}

We can even combine property 1 and 2 to show that:

T(cu+dv)=cT(u)+dT(v)T(cu+dv)=cT(u)+dT(v)

where uu, vv are vectors and cc, dd are scalars. Note that if this equation holds, then it must be linear.

If you have more than 2 vectors and 2 scalars? What if you have p vectors and p scalars? Then we can generalize this equation and say that:

T(c1v1+c2v2++cpvp)=c1T(v1)+c2T(v2)++cpT(vp) T(c_1 v_1+c_2 v_2+\cdots+c_p v_p )=c_1 T(v_1 )+c_2 T(v_2 )+\cdots+c_p T(v_p)

Again if this equation holds, then it must be linear.
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Properties of linear transformation

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