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If \(\displaystyle T:V \to V\) is a linear transformation, and \(\displaystyle A\) is a matrix repredenting \(\displaystyle T\) with respect to the basis \(\displaystyle (e_1, e_2, \cdots e_n)\) and \(\displaystyle B\) is a matrix representing the same linear transformation \(\displaystyle T\) with respect to basis \(\displaystyle (u_1, u_2, \cdots u_n)\). Then, I want to ask what will be the relation between effects of these matrices on an element of \(\displaystyle V\). Let’s say \(\displaystyle T(x)=y\). Well, then we should have $$ A ~x = y = B ~x $$ But is that the case? I don’t think two different matrices multplied by same column vector will give out the same output.
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Adesh said: If \(\displaystyle T:V \to V\) is a linear transformation, and \(\displaystyle A\) is a matrix repredenting \(\displaystyle T\) with respect to the basis \(\displaystyle (e_1, e_2, \cdots e_n)\) and \(\displaystyle B\) is a matrix representing the same linear transformation \(\displaystyle T\) with respect to basis \(\displaystyle (u_1, u_2, \cdots u_n)\). Then, I want to ask what will be the relation between effects of these matrices on an element of \(\displaystyle V\). Let’s say \(\displaystyle T(x)=y\). Well, then we should have $$ A ~x = y = B ~x $$ But is that the case? I don’t think two different matrices multplied by same column vector will give out the same output. Click to expand... In order to make any sense of Tx = y T and the vectors need to be written in the same basis. So if you use a different basis to represent T, then you have T'x' = y'... the coefficients are all different. So whereas y and y' represent the same vector they will look different. Geometrically speaking y will be the same as y' but the coordinate systems will be different. -Dan
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