1. anonymous

Still here?

2. anonymous

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.

3. anonymous

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.

4. anonymous