# Equivalent matrices represent the same linear transformation with resect to appropriate bases.

Put content here**Theorem:** Equivalent matrices represent the same linear transformation with respect to appropriate bases.
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**Explanation:** If $A$ and $B$ are equivalent matrices (i.e., $B = PAQ$ for invertible $P$ and $Q$), then there exist bases $\alpha, \beta$ such that $A = [T]_{\alpha}^{\beta}$ and $B = [T]_{\alpha}^{eta}$ for the same linear transformation $T$.
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**Connection:** Matrix equivalence captures the idea that the same abstract linear map looks different when expressed in different coordinate systems. The change-of-basis matrices $P$ and $Q$ account for the coordinate transformations in the domain and codomain.

# Parents

* Matrix equivalence