Properties of linear transformation

Lessons

• Introduction
  Properties of Linear Transformation Overview:
  a)

  The 3 properties of Linear Transformation
  • $T(u+v)=T(u)+T(v)$
  • $T(cu)=cT(u)$
  • $T(0)=0$
  
  
  b)

  How to see if a transformation is linear
  • Show that: $T(cu+dv)=cT(u)+dT(v)$
  • General formula: $T(c_1 v_1+c_2 v_2+\cdots+c_n v_n )=c_1 T(v_1 )+c_2 T(v_2 )+\cdots+c_p T(v_p)$
  
  

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• 1.
  Understanding and Using the Properties
  Show that the transformation $T$ defined by is not linear.

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• 2.
  Show that the transformation $T$ defined by is not linear.

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• 3.
  Proving Questions using the Properties
  An affine transformation $T: \Bbb{R}^n$→$\Bbb{R}^m$ has the form $T(x)=Ax+b$, where $A$ is an $m \times n$ matrix and $b$ is a vector in $\Bbb{R}^n$. Show that the transformation $T$ is not a linear transformation when $b \neq 0$.

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• 4.
  Define $T: \Bbb{R}^n$→$\Bbb{R}^m$ to be a linear transformation, and let the set of vectors {$v_1,v_2,v_3$ } be linearly dependent. Show that the set of vectors {$T(v_1),T(v_2),T(v_3)$} are also linearly dependent.

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• 5.
  Define $T: \Bbb{R}^n$→$\Bbb{R}^m$ to be a linear transformation and the set of vectors $v_1$,...,$v_p$ are in $\Bbb{R}^n$. In addition, let $T(v_i )=0$ for $i=1,2,$…$,p$. If $x$ is any vector in $\Bbb{R}^n$, then show that $T(x)=0$. In other words, show that $T$ is the zero transformation.

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