Hi: can a linear transformation have the same matrix in different bases? I have spent a whole evening trying to figure out whether this can be to no use. What is certain is that given a base, I can define a linear ttransformation by giving it's values over a base. That is, there exists a function theta from the set of all ordered bases of V to the set of all linear transformations. Does this function have an inverse?
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- 4$\begingroup$ Cheap example: the linear operator which sends everything to zero has the same matrix for each choice of basis. $\endgroup$D_S– D_S2016-08-07 22:55:25 +00:00Commented Aug 7, 2016 at 22:55
- 5$\begingroup$ so does the mapping that sends every vector to itself $\endgroup$Will Jagy– Will Jagy2016-08-07 22:56:09 +00:00Commented Aug 7, 2016 at 22:56
- 1$\begingroup$ "That is, there exists a function theta from the set of all ordered bases of V to the set of all linear transformations." What you are describing is not function from the set of ordered bases to the set of linear transformations. A function must have exactly one output for each input. $\endgroup$D_S– D_S2016-08-07 23:01:21 +00:00Commented Aug 7, 2016 at 23:01
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1 Saying that a linear operator $f$ has the same matrix $A$ in 2 different bases $B$ and $B'$ means that, if $P$ is the matrix of the coordinates of the vectors of $B'$ relatively to $B$, $$A=PAP^{-1}$$ Hence $$AP=PA$$ So it may be possible that a same matrix represent $f$ in two different bases.
- $\begingroup$ Thanks Paf. Now suppose this is (V,f), the set of all alternate forms over V x V. Then to f there corresponds a sympletic basis, and the values of f are determined by that base, that is, the matrix of f over the base consists of 2 x 2 square blocks with zeroes in the main d iagonal and +/-1 on the other. Can f have the same matrix in another basis. @D_S: if you assign values to each element of a basis you get a linear trasformation, for the value at each element of the domain will be a linear combination of the values at the elements of the base. Am I wrong? Hence, you get a function from the $\endgroup$stf92– stf922016-08-08 01:41:24 +00:00Commented Aug 8, 2016 at 1:41
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Here is a slightly less trivial example (than the zero and identity transformations), which is easily generalized.
Let $T:\mathbb{R}^2\to \mathbb{R}^2$ be the linear transformation given by $T(a,b)=(a+3b,2a+4b)$ and let $\beta$ be the standard ordered basis and $\gamma$ be the ordered basis $\gamma=\{(1,2), (3,4)\}$. Then $$[T]_\beta=\begin{bmatrix} 1 & 3\\ 2 & 4 \end{bmatrix} =[T]_\gamma$$
