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December 15th, 2017, 09:55 AM  #1 
Newbie Joined: Dec 2017 From: Cosenza Italy Posts: 5 Thanks: 0  Linear application and matrix.
Hello everyone, I have a doubt. I know that the image of a generic domain vector, defined by the linear application, is equivalent to the vectormatrix product between the matrix associated with the application and a generic domain vector. Is the associated matrix that we use always built with canonical bases or can you use associated arrays built with a generic base of the domain and a generic base of the codomain? Thank you all

December 15th, 2017, 11:32 AM  #2 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 896 
Given any basis, not just "canonical" bases, there exist a matrix representing the linear transformation in that basis. In fact, you can have one basis for the "domain" space and another for the range space. For example, suppose T is the linear transformation that maps (x, y, z) into (3x+ y, y z, x+ y+ z). Take the basis for the domain space to be {(1, 0, 1), (1, 1, 0), (0, 1, 1)} and the basis for the range space to be {(1, 0, 0), (1, 1, 1), (3, 2, 1)}. Apply T to (1, 0, 1). That gives (3+ 0,0 1, 1+ 0+ 1)= (3, 1, 2). Write that as a linear combination of {(1, 0, 0), (1, 1, 1), (3, 2, 1)}: (3, 1, 2)= a(1, 0, 0)+ b(1, 1, 1)+ c(3, 2, 1)= (a+ b+ 3c, b+ 2c, b+ c). That gives the three equations a+ b+ 3c= 3, b+ 2c= 1, and b+ c= 2. Subtracting the last equation from the second, c= 3. Then b+ (3)= 2 so b= 5. a+ b+ 3c= a+ 5 3= a+ 2= 3 so a= 1. The first column of the matrix representation of T, in those bases, is $\displaystyle \begin{pmatrix}1 \\ 5 \\ 3\end{pmatrix}$. Apply T to (1, 1, 0) and write the result as a linear combination of {(1, 0, 0), (1, 1, 1), (3, 2, 1)} to get the second column and to (0, 1, 1) to get the third column. 
December 15th, 2017, 12:31 PM  #3 
Newbie Joined: Dec 2017 From: Cosenza Italy Posts: 5 Thanks: 0 
Hello Country Boy, yes i know how to built a matrix associated to a linear application. I mean this: T is a linear application T: R^2>R^3 1 2 is the matrix associated to this application 4 7 7 8 (canonical bases) I can calculate the image of a generic vector of the domain through the vector product matrix between the representative matrix and a generic domain vector. 1 2 4 7 (x,y) = (x+2y,4x+5y,7x+8y) 7 8 For example T(2,1) = (4,13,24) But if I make this product with a representative matrix constructed from noncanonical bases then the image of this same vector will be different from the previous one. So I do not know if I should report (to define the image of a generic vector of the domain) only to matrixes built in relation to the canonical bases. Thank you very much.Sorry for my english. Last edited by JackPirri; December 15th, 2017 at 12:36 PM. 

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