My Math Forum Linear application and matrix.

 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 Display Modes Linear 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

 Contact - Home - Forums - Cryptocurrency Forum - Top