Properties of linear transformation

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Intros
Lessons
  1. Properties of Linear Transformation Overview:
  2. The 3 properties of Linear Transformation
    T(u+v)=T(u)+T(v)T(u+v)=T(u)+T(v)
    T(cu)=cT(u)T(cu)=cT(u)
    T(0)=0T(0)=0
  3. 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)
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Examples
Lessons
  1. Understanding and Using the Properties
    Show that the transformation TT defined by understand properties of linear transformation is not linear.
    1. Show that the transformation TT defined by prove linear transformation is not linear.
      1. 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.
        1. 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.
          1. 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.