0 like 0 dislike
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.

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