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Homework Statement:

Is the following statement true or false? If it is the former case, prove it. If it is later, give a counterexample.
Let ##n \in \aleph_0## and ##L:\Re^{n} \rightarrow \Re^{n}## be an injective linear mapping. Let ##A \in \Re^{n \times n}## be an invertible matrix. Then there is a basis ##\alpha## of ##\Re^{n}## and a basis ##\beta## of ##\Re^{n}## such that ##A = L_{\alpha}^{\beta}##
Homework Equations:
 Please check out diagram
I know that to go from a vector with coordinates relative to a basis ##\alpha## to a vector with coordinates relative to a basis ##\beta## we can use the matrix representation of the identity transformation: ##\Big( Id \Big)_{\alpha}^{\beta}##.
This can be represented by a diagram:
Thus note that the linear mapping we are interested in is ##A:X \rightarrow X'##, where:
$$A = \Big( Id \Big)_{\alpha}^{\beta}$$
I think that the statement is true but I think I should use the fact that ##A## is invertible somehow on the above equation in order to prove it. But how?
This can be represented by a diagram:
Thus note that the linear mapping we are interested in is ##A:X \rightarrow X'##, where:
$$A = \Big( Id \Big)_{\alpha}^{\beta}$$
I think that the statement is true but I think I should use the fact that ##A## is invertible somehow on the above equation in order to prove it. But how?
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