I was just wondering if someone could help me with some of theterminology used i

Question

I was just wondering if someone could help me with some of theterminology used in liinear. I keep reading and reading thetextbook and I seem to get more confused all the time. I want to start from way back to see I can get a grip on whatis going on Can someone explain linear transformation and what isgoing on when we do a transformation? The book did a chapter on determinants and I have a decenthandle on them but what do we use them for? We then all of the sudden jumped or it seemed into dot products and crossproducts which I think I can do when do we use thesetechniques. I have so many questions but I will only ask one morehere. When we do transformations what are we accomplishingwhen we do this? I guess I am missing the whole point of thesubject. I was just wondering if someone could help me with some of theterminology used in liinear. I keep reading and reading thetextbook and I seem to get more confused all the time. I want to start from way back to see I can get a grip on whatis going on Can someone explain linear transformation and what isgoing on when we do a transformation? The book did a chapter on determinants and I have a decenthandle on them but what do we use them for? We then all of the sudden jumped or it seemed into dot products and crossproducts which I think I can do when do we use thesetechniques. I have so many questions but I will only ask one morehere. When we do transformations what are we accomplishingwhen we do this? I guess I am missing the whole point of thesubject.

Explanation / Answer

dear ! i am here to take you through all the way about lineartransformation. don't worry. ofcourse , you mention the names of each concept ortheorem. i will try my best to make you understand. o.k. ? now, a linear transformation is a function, homomorphism from a vector space to a vector space. no square or cube or higher terms are allowed except degreeone. that is why we use the word linear. followed ? now, to establish the given function T : U--> V to be alinear transformation, we are required to show T ( au + bv) = aT(u)+bT(v) for any given vectors u , v in U and the scalars a ,b. for instance , T : R3 -->R2 defined by T( x,y,z) = ( x+y , y - z) is a lineartransformation. consider two vectors in R3 . i.e. u = (x1,x2,x3) and v = ( y1,y2,y3) and a , b are anyscalars. consider au + bv = a( x1,x2,x3)+b(y1,y2,y3) = (ax1,ax2,ax3) + ( by1,by2,by3) = (ax1+by1,ax2+by2,ax3+by3)      applying T on this. i.e.T(au+bv) = T(ax1+by1,ax2+by2,ax3+by3)                                            = ((ax1+by1)+(ax2+by2) , (ax2+by2)-(ax3+by3))   bydefinition of T.                                 = ( a(x1+x2) +b(y1+y2), a(x2-x3) +b(y2-y3))                                   = ( a(x1+x2), a(x2-x3)) + (b(y1+y2), b(y2-y3))                                 = a(x1+x2,x2-x3) + b(y1+y2, y2-y3)                                 = aT( x1,x2,x3)+bT ( y1,y2,y3)                                 = aT(u) +bT(v) so, the definition of linear transformation issatisfied. so, T is a linear transformation.

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