Question

what is linear transformation in matrixes please help
a simple example please

Answer 1

Still here?

Answer 2

If you let A be a m x n matrix, then the function T defined by \[T(v)=Av\] is a linear transformation from R^n into R^m (i.e v is a vector in R^n). Basically, multiplying this matrix by these vectors is a linear transformation because:\[T(v+w)=A(v+w)=Av+Aw=T(v)+T(w)\]for v,w in R^n and\[T(\lambda v) = A(\lambda v)=\lambda Av=\lambda T(v)\]where lambda is a scalar.

So, matrices can take vectors from one space to another in a linear way.

Answer 3

A common example of linear transformation in matrices is the rotation matrix:

A=\ cos theta -sin theta\
\sin theta cos theta\

This is a linear transformation on 2-dimensional vectors, like, v=(2,1) and w=(3,4) since:
T(v+w)=
A[(2,1)+(3,4)]=A[(5,5)]= \5cos theta -5sin theta\
\5sin theta + 5cos theta\ ...(1)
and
T(v)+T(w)=
A[(2,1)] + A[(3,4)] = \2cos theta -sin theta\ + \3cos theta - 4sin theta\
\2sin theta + cos theta\ \3sin theta + 4cos theta\

=\5cos theta -5sin theta\
\5sin theta +5cos theta\ ...(2)

i.e. T(v+w)=T(v)+T(w)

You can show the scalar condition as well by doing something similar (i.e. evaluating T(lambda v) = A(lambda (1,2)), and
(lambda)T(v) = (lambda)A(1,2)

and showing the two are equal.

I apologize if this confuses matters...it's *really* difficult to communicate everything online. Let me know if you need more help.

Answer 4

Some sites:

http://www.youtube.com/results?search_query=linear+transformation+examples&aq=1&oq=linear+trans

http://www.youtube.com/results?search_query=linear+transformation+matrices&aq=0&oq=linear+transformation+mat