Linear Algebra 19f: The Heart of Component Spaces - Matrix Representation of a Linear Tranformation

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[@MathTheBeautiful]

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[@GSOG2]

This one video finally helped me understand how to build the transformation matrix. No other book or video got it passed my thick skull.

[@cardinalblues7121]

I am 40 yo now ，I finally tasted the beauty of maths. I am eager to learn from you, thank you

[@MathTheBeautiful]

I'm so glad to hear it!

[@thentust]

Why the components of the images of the basis vectors times the vector V could transform to R(V)?

[@MathTheBeautiful]

+thentust What part of the video are you referring to? Maybe you could write some expressions to make your question clearer?

[@thentust]

+MathTheBeautiful
I see the transform matrix is the components of the images of the basis vectors as you said, but I dont know why the matrix already have a transform function and makes any arbitrary vector times it could be transform of reflection?

[@MathTheBeautiful]

+thentust The argument goes like this:
T(v) = T(a1 e1 + a2 e2 + a3 e3) = a1 T(e1) + a2 T(e2) + a3 T(e3)
Since the vectors T(e1), T(e2), and T(e3) are correctly represented by the columns of the matrix, T(v) is represented by the corresponding linear combination of the those columns. Meanwhile, this correct linear combination can be expressed by the matrix product A times [ a1 a2 a3 ]. So the method works.