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 Linear Algebra Linear Algebra Math Forum

 December 15th, 2017, 08:55 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 vector-matrix 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, 10:32 AM #2 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 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, 11:31 AM #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 non-canonical 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 11:36 AM. Tags application, linear, matrix Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post nhinhiaus Linear Algebra 1 May 19th, 2016 04:27 AM Gaz41 Real Analysis 1 March 21st, 2016 09:59 PM richardz03 Number Theory 0 January 14th, 2015 04:59 PM david940 Linear Algebra 3 June 29th, 2014 02:45 AM remeday86 Linear Algebra 1 August 14th, 2010 10:37 PM

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