How to Find the Matrix for a Linear Transformation Relative to Standard Bases

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Intro Linear Algebra
How to find the matrix for a linear transformation from P2 to R3, relative to the standard bases for each vector space. The same technique applies in general, and for non-standard bases, although the steps are more involved.
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Thank you so much! I was really struggeling to understand how to solve this kind of equation. You're a lifesaver! :)

Nini-vwzn
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How can you apply this method when the second basis contains more elements than the first one?

coolmaster-ghtr