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Properties of linear transformation
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Properties of linear transformation
Lessons
Recall from last chapter the 2 properties of Ax:
1. A(u+v)=Au+Av
2. A(cu)=c(Au)
where u and v are vectors in Rn and c 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)
2. T(cu)=cT(u)
3. T(0)=0
We can even combine property 1 and 2 to show that:
T(cu+dv)=cT(u)+dT(v)
where u, v are vectors and c, d 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)
Again if this equation holds, then it must be linear.
1. A(u+v)=Au+Av
2. A(cu)=c(Au)
where u and v are vectors in Rn and c 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)
2. T(cu)=cT(u)
3. T(0)=0
We can even combine property 1 and 2 to show that:
where u, v are vectors and c, d 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:
Again if this equation holds, then it must be linear.
- IntroductionProperties of Linear Transformation Overview:a)The 3 properties of Linear Transformation
• T(u+v)=T(u)+T(v)
• T(cu)=cT(u)
• T(0)=0b)How to see if a transformation is linear
• Show that: T(cu+dv)=cT(u)+dT(v)
• General formula: T(c1v1+c2v2+⋯+cnvn)=c1T(v1)+c2T(v2)+⋯+cpT(vp) - 1.Understanding and Using the Properties
Show that the transformation T defined byis not linear.
- 2.Show that the transformation T defined by
is not linear.
- 3.Proving Questions using the Properties
An affine transformation T:Rn→Rm has the form T(x)=Ax+b, where A is an m×n matrix and b is a vector in Rn. Show that the transformation T is not a linear transformation when b≠0. - 4.Define T:Rn→Rm to be a linear transformation, and let the set of vectors {v1,v2,v3 } be linearly dependent. Show that the set of vectors {T(v1),T(v2),T(v3)} are also linearly dependent.
- 5.Define T:Rn→Rm to be a linear transformation and the set of vectors v1,...,vp are in Rn. In addition, let T(vi)=0 for i=1,2,…,p. If x is any vector in Rn, then show that T(x)=0. In other words, show that T is the zero transformation.