Properties of linear transformation

Properties of linear transformation


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:


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.
  • 1.
    Properties of Linear Transformation Overview:
    The 3 properties of Linear Transformation

    How to see if a transformation is linear
    • Show that: T(cu+dv)=cT(u)+dT(v)T(cu+dv)=cT(u)+dT(v)
    • General formula: T(c1v1+c2v2++cnvn)=c1T(v1)+c2T(v2)++cpT(vp)T(c_1 v_1+c_2 v_2+\cdots+c_n v_n )=c_1 T(v_1 )+c_2 T(v_2 )+\cdots+c_p T(v_p)

  • 2.
    Understanding and Using the Properties
    Show that the transformation TT defined by understand properties of linear transformation is not linear.

  • 3.
    Show that the transformation TT defined by prove linear transformation is not linear.

  • 4.
    Proving Questions using the Properties
    An affine transformation T:RnT: \Bbb{R}^n Rm \Bbb{R}^m has the form T(x)=Ax+bT(x)=Ax+b, where AA is an m×nm \times n matrix and bb is a vector in Rn\Bbb{R}^n. Show that the transformation TT is not a linear transformation when b0b \neq 0.

  • 5.
    Define T:RnT: \Bbb{R}^n Rm \Bbb{R}^m to be a linear transformation, and let the set of vectors {v1,v2,v3v_1,v_2,v_3 } be linearly dependent. Show that the set of vectors {T(v1),T(v2),T(v3)T(v_1),T(v_2),T(v_3)} are also linearly dependent.

  • 6.
    Define T:RnT: \Bbb{R}^n Rm \Bbb{R}^m to be a linear transformation and the set of vectors v1v_1,...,vpv_p are in Rn\Bbb{R}^n. In addition, let T(vi)=0T(v_i )=0 for i=1,2,i=1,2,,p,p. If xx is any vector in Rn\Bbb{R}^n, then show that T(x)=0T(x)=0. In other words, show that TT is the zero transformation.